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Elements of surgery theoryR U S TA M S A DY K O V

E L E M E N T S O F S U R G E R Y T H E O R Y

Contents

1 Stable homotopy groups of spheres 15

1.1 Framed submanifolds in Euclidean spaces 16

1.2 Cobordisms of framed manifolds 18

1.3 The Pontryagin construction 20

1.4 Example: The stable group πS 0 22

1.5 Further reading 23

2 Immersion Theory 24

2.1 Ambient isotopy 25

2.4 Further reading 30

3 Spherical surgeries 32

3.2 Framed base of surgery 37

3.3 Reduction of homotopy groups 38

3.4 Examples: Stable homotopy groups πS 1 and πS

2 39

4.1 The Whitney weak embedding and immersion theorems 45

4.2 The Whitney trick 46

4.3 Strong Whitney embedding theorem 50

4.4 Strong Whitney immersion theorem 51

4.5 Solution to Exercises 53

4.6 Further reading 53

5.1 The Wall representative and the invariant µ 56

5.2 Homotopy spheres 61

5.4 Main theorems 66

5.5 Further reading 67

6.1 Invariance of signature and Kervaire invariant 71

6.2 The main theorems for even dimensional manifolds 73

6.3 The main theorems for odd dimensional manifolds 76

6.4 Solutions to exercises 79

7 Characteristic classes of vector bundles 84

7.1 Vector bundles over topological spaces 84

7.2 Characteristic classes of vector bundles 85

7.2.1 The Euler class 86

7.2.2 Stiefel-Whitney classes 87

7.2.3 Chern classes 89

7.2.4 Pontryagin classes 89

8.1 Maps of SU(2) and Spin(4) to orthogonal groups 94

8.1.1 The map SU(2)→ SO(3) 94

8.1.2 The map Spin(4)→ SO(4) 94

8.2 Milnor’s fiber bundles 95

4

8.4 Invariants of Milnor’s fibrations 98

8.5 Exotic spheres of dimension 7 100

8.6 Further reading 100

9 Exotic spheres 102

9.1 The group of smooth structures on a sphere 102

9.2 The surgery long exact sequence for spheres 105

9.3 Calculation of the group Pm 106

9.4 Calculation of the group Nm. 108

9.5 Calculation of homomorphisms in the surgery exact sequence 110

9.6 Hirzebruch genera and computation of the L-genus 111

9.6.1 Multiplicative series 111

9.6.2 Genera of manifolds M4n with trivial p1, ..., pn−1. 113

9.7 Example: The third stable homotopy group πs 3 114

9.8 Further reading 115

10.1.1 Handle rearrangement 117

10.1.2 Handle creation 118

10.1.3 Handle cancellation 119

10.1.4 Handle slides 119

10.1.5 Handle trading 120

11 Surgery on maps of simply connected manifolds 127

11.1 Normal maps 127

11.2.1 Tensor products 130

11.2.3 Poincaré Duality 132

12 The surgery long exact sequence 137

5

13.1 The Whitney trick in non-simply connected manifolds 142

13.2 Right and left Modules 144

13.3 Homology and cohomology with twisted coefficients 145

13.4 Products in (co)homology with twisted coefficients 147

13.5 Poincaré complexes 148

13.6 Homological algebra 150

14.2 Surgery on maps of manifolds of even dimension 154

14.2.1 Wall representatives 154

14.2.2 The intersection pairing λ and self-intersection form µ 156

14.2.3 Effect of surgery on homotopy groups 159

14.2.4 The main theorem 159

14.3 Surgery on maps of manifolds of odd dimension 160

14.3.1 Heegaard splittings 160

14.3.2 ε-quadratic formations 163

14.4 Further reading 168

99 Additional Topics 171

99.1.1 Hopf fibration 172

99.1.3 The Pontryagin-Thom construction 174

99.1.4 The stable J-homomorphism 175

99.2 Topics for Chapter 2 177

99.2.1 The Multicompression theorem 177

99.2.2 The Smale-Hirsch Theorem 179

99.2.3 The Smale Paradox 181

99.2.4 Cobordisms of oriented immersed manifolds 182

99.2.5 The Local Compression Theorem 183

99.2.6 Solutions to Exercises 186

99.3 Topics for Chapter 3 189

99.3.1 Example: Cobordisms 189

6

99.4.1 Bilinear forms on free modules 194

99.5 Topics for Chapter 6 195

99.5.1 Heegaard splittings 195

99.5.2 Alternative proof of the surgery theorem for framed manifolds of odd dimension 196

99.6 Topics for Chapter 14 199

99.6.1 The Whitehead torsion 199

99.6.2 The s-cobordism theorem 200

Figure 1: A CW-complex is constructed by induction beginning with a (red) dis- crete set X0, by attaching (blue) seg- ments, then (grey) 2-dimensional cells and so on.

0

0.1 Homotopy groups

Pointed topological spaces. A pointed topological space is a topological space X together with a point ∗ ∈ X called the basepoint of X. A map of pointed topological spaces is required to take the basepoint in the domain to the basepoint in the target. The pointed union X ∨ Y of a pointed topological spaces X and Y is the space X tY/∼ in which the basepoints of X and Y are identified.

Example 0.1. Let Sn denote the standard unit sphere centered at ori- gin in the Euclidean space Rn+1 with basis e1, ..., en. The sphere Sn be- comes pointed if we pick its south pole, i.e., the end point of −en+1 to be the basepoint of Sn. The intersection of Sn with the hyperplane per- pendicular to e1 is a meridian Sn−1 of Sn. The quotient space Sn/Sn−1

is homeomorphic to the bouquet of spheres Sn ∨ Sn.

A homotopy of a map f : X → Y to a map g : X → Y is a continuous family of maps Ft : X → Y parametrized by t ∈ [0, 1] such that F0 = f and F1 = g. When the spaces X and Y are pointed, every map Ft in the homotopy is required to be pointed as well. A space X is homotopy equivalent to a space Y if there are maps f : X → Y and g : Y → X such that g f is homotopic to the identity map of X, and f g is homotopy equivalent to the identity map of Y.

CW complexes. Most topological spaces that we will encounter are CW complexes. A CW complex X is constructed inductively beginning with a discrete set X0 called the 0-skeleton of X. The n-skeleton Xn

is obtained from the (n − 1)-skeleton Xn−1 by attaching a collection

8 elements of surgery theory

tDn α of closed discs of dimension n with respect to attaching maps

α : ∂Dn α → Xn−1. In other words, Xn consists of Xn−1 and the discs

Dn α with each x in the boundary ∂Dn

α identified with α(x) ∈ Xn−1. We say that the interior of ∂Dn

α is an open cell en α . The so-constructed

space is a CW-complex X = ∪Xn. A set U in X is declared to be open (respectively, closed) if each intersection U ∩ Xn is open (respectively, closed).

Homotopy groups. For n ≥ 0, the set of homotopy classes [ f ] of pointed maps f : Sn → X into a pointed topological space is denoted by πnX. The set πnX is a group when n > 0 with the sum [ f ] + [g] represented by the composition

Sn −→ Sn/Sn−1 ≈ Sn ∨ Sn f∨g−−→ X,

where the first map is one collapsing the meridian of Sn into a point. The set π0X is the set of path components of X. The group π1X is the fundamental group. The homotopy groups πnX with n > 1 are abelian. A space X is n-connected if the groups πiX are trivial in the range i ≤ n. Thus, a non-empty space is −1-connected, a path connected space is 0-connected, while a simply connected space is 1-connected. A map f : X → Y of topological spaces induces a homomorphism f∗ : πiX → πiY of homotopy groups by [g] 7→ [ f g].

Theorem 0.2 (Whitehead). Two path-connected CW-complexes X and Y are homotopy equivalent if and only if there exists a map f : X → Y which induces an isomorphism of homotopy groups f∗ : πiX → πiY for all i ≥ 0.

Relative homotopy groups. A pair (X, A) of pointed topological spaces consists of a pointed topological space X and a pointed subspace A such that the distinguished point of A is the same as the distinguished point of X. A map of pairs (X, A) → (Y, B) is a pointed map from X to Y which takes the subspace A to B. Similarly, a homotopy of maps f0, f1 of pairs is a continuous family Ft of maps of pairs parametrized by t ∈ [0, 1] such that F0 = f0 and F1 = f1. Let Dn denote the stan- dard disc of dimension n, and Sn−1 its boundary. Then, the set of homotopy classes of maps from the pair (Dn, Sn−1) to a pair (X, A) is denoted by πn(X, A). It is a group if n ≥ 2, and an abelian group for n ≥ 3. We note that when the subspace A is a point, the relative ho- motopy groups reduce to homotopy groups: πn(X, ∗) = πnX. Every map of pairs f : (X, A) → (Y, B) induces a map f∗ of relative homo- topy groups π∗(X, A)→ π∗(Y, B). For any pair (X, A), there is a long exact sequence of homotopy groups:

· · · → πn+1(X, A) ∂−→ πn A i∗−→ πnX

j∗−→ πn(X, A)→ · · · ,

background information 9

where i : A → X is the inclusion, j is the inclusion (X, ∗) ⊂ (X, A)

of pairs, while ∂ takes the class [ f ] of a map f : Dn → X to the class [ f |∂Dn] of the map f |∂Dn : Sn−1 → A.

A pair (X, A) is said to be n-connected if πi(X, A) = 0 for i ≤ n and any choice of a base point ∗ ∈ A.

Theorem 0.3. Let X be a CW complex decomposed as the union of sub- complexes A and B with nonempty connected intersection C = A ∪ B. If (A, C) is m-connected and (B, C) is n-connected, m, n ≥ 0, then the map πi(A, C)→ πi(X, B) induced by inclusion is an isomorphism for i < m + n and a surjection for i = m + n.

Theorem 0.4 (Hatcher, Proposition 4.28). If a CW pair (X, A) is r-connected and A is s-connected, with r, s ≥ 0, then the map πi(X, A) → πi(X/A)

induced by the quotient map X → X/A is an isomorphism for i ≤ r + s and a surjection for i = r + s + 1.

Let A and B be two vector spaces over a field k, or finitely gener- ated free abelian groups (in which case we put define k to be the ring Z). We say that a bilinear function ψ : A ⊗ B → k is non-degenerate with respect to A if for any linear function f on A there is a unique vector b ∈ B such that f (a) = ψ(a, b). Similarly, we say that ψ is non- degenerate with respect to the second factor if for any function f on B there is a vector a ∈ A such that f (b) = ψ(a, b).

Theorem 0.5. Suppose that W is an oriented compact manifold of dimension 2q + 1 ≥ 5. Suppose that W and ∂W are q− 1 connected manifolds. Then the intersection pairing

πqW/Tor⊗ πq+1(W, ∂W)/Tor → Z

is non-degenerate with respect to the first factor, while the pairing

πq+1W/Tor⊗ πq(W, ∂W)/Tor → Z

is non-degenerate with respect to the second factor.

Proof. Let us prove the non-degeneracy of the first pairing. For the second pairing, the proof is similar.

By the Hurewicz isomorphism, the free part of πqW is isomorphic to the free part of HqW. Since the homology pairing is non-degenerate, for every function f on the free part of HqW there is an element x f in the free part of Hq+1(W, ∂W) such that f (y) = x f · y for all elements y

10 elements of surgery theory

1 See Exercise 23 in section 4.2 of Hatcher’s book.

in the free part of Hq+1(W, ∂W). By the Hurewicz theorem, the homo- morphism πq+1(W/∂W) → Hq+1(W/∂W) is surjective.1 Therefore, the element x f lifts to an element in πq+1(W/∂W). Finally, x f admits a lift with respect to the map πq+1(W, ∂W) → Hq+1(W/∂W) since such a map is surjective by Theorem 0.4.

For the second pairing, we have πq(W, ∂W) ≈ πq(W/∂W) is isomor- phic to Hq(W, ∂W). Therefore, for any function f on the part of this group, there is a corresponding element x f in Hq+1(W) which can be lifted to an element in πq+1W.

0.2 Bordisms and Cobordisms

Bordisms. We say that two maps f0 : M0 → X and f1 : M1 → X of closed oriented manifolds into a topological space are bordant, if there is a map f : W → X of a compact manifold W with boundary ∂W = M0 t (−M1) such that f |M0 = f0 and f |M1 = f1. The set of bordism classes of maps of manifolds of dimension m into a manifold N forms a group mX with operation given by taking the disjoint union of maps.

More generally, given a compact oriented manifold M, a continuous map f : (M, ∂M) → (X, A) is bordant to zero if there is a compact ori- ented manifold W with boundary ∂W = MtM′ and a map F : W → X such that F|M = f and F(M′) ⊂ A. Two maps fi : (Mi, ∂Mi)→ (X, A)

are said to be bordant if f0 t f1 : (M0 ∪−M1, ∂M0 ∪−∂M1)→ (X, A) is bordant to zero. The set of bordant maps of manifolds of dimension m forms an abelian group m(X, A). Clearly, we have m(X, ∅) = mX.

Every map g : (X′, A′) → (X, A) of pairs induces a homomorphism g∗ : ∗(X′, A′) → ∗(X, A) by [ f ] 7→ [g f ]. Homotopic maps g, g′

induce the same homomorphism g∗ = g′∗. There is a boundary ho- momorphism ∂ : m(X, A) → m−1 A defined by associating the class of f : (M, ∂M) → (X, A) with the class f |∂M : ∂M → A. For any pair (X, A) of spaces, there is a long exact sequence of abelian groups

→ m+1(X, A) ∂−→ m A→ mX → m(X, A)→,

where i : A → X and j : (X, ∅) → (X, A) are the inclusions. The re- duced bordism group mX is the kernel of the homomorphism ε∗ in- duced by the projection ε of X onto a one point space. There is an iso- morphism m(X, A) ≈ m(X/A) where X/∅ is the disjoint union of

background information 11

2 Note that oriented bordism classes are represented by maps of oriented closed manifolds, while oriented cobor- dism classes are represented by oriented proper maps of manifolds which are not necessarily oriented or closed.

X and a one point set. Suppose that U is a subset in X such that its clo- sure is in the interior of A. Then the inclusion (X \U, A \U)→ (X, A)

induces an isomorphism of bordism groups.

Cobordisms. We say that a proper map f : M→ N of manifolds is ori- ented if the normal bundle of a map f × i : M→ N×Rk for an embed- ding i : M → Rk with k 1 is oriented. Two orientations defined by i and i′ are equivalent, if they are compatible with an isotopy from i to i′. We say that two proper oriented maps f0 : M0 → N and f1 : M0 → N are cobordant if there is a proper oriented map f : W → N of a manifold with boundary ∂W = M0 ∪M1 extending f0 and f1 in such a way that the orientation of f agrees with the orientations of f0 and f1. The set of cobordism classes of maps of dimension −q = dim M− dim N into N is a group denoted by qN. 2

Every smooth map g : N′ → N of manifolds defines a homomorphism g∗ : ∗N → ∗N′ by [ f ] 7→ [g∗ f ]. The homomorphism g∗ depend only on the homotopy class of g. A proper oriented map g : N′ → N of dimension d defines the so-called Gysin homomorphism g! : mN′ → m−dN by [ f ] 7→ [g f ].

Products. The external products

∧ : mN ⊗m′N ′ → m+m′(N × N′),

∧ : dN ⊗d′N ′ → d+d′(N × N′)

are defined by [ f1] ∧ [ f2] = [ f1 × f2]. There are also homomorphisms

/ : q(N × N′)⊗mN′ → q−m(N),

\ : qN ⊗m(N × N′)→ m−q(N′).

The first homomorphism is defined by [ f ]/[g] = [(π1 f f ∗(1N × g)]. For example, suppose that both f and g are transverse embeddings. Then f f ∗(1N × g) is the inclusion into N × N′ of the intersection of the images of f and g. The map π1 projects it to N. The second homomorphism is defined by [ f ]\[g] = [π2 g g∗( f × 1N′)].

The group ∗N is a ring with operation given by [ f1] ∪ [ f2] = ∗( f1 ∧ f2) where : N → N × N is the diagonal map. There is also the ∩- product

∩ : qN ⊗m(N, ∂N) −→ m−q(N, ∂N),

defined by the composition

\−→ m−q(N, ∂N).

12 elements of surgery theory

When N is a compact orientable manifold of dimension n, the class [N] of the identity map of (N, ∂N) is called the fundamental class of N in n(N, ∂N).

0.3 Homology and Cohomology

Let X be a CW-complex. We recall that it is constructed by induction beginning with a discrete set X0. The n-th skeleton Xn is obtained from the (n− 1)-skeleton by attaching to Xn−1 closed discs Dn

α of dimension n by means of the attaching maps α : ∂Dn

α → Xn−1. The interior en α of

the attached disc Dn α is called an open cell.

The algebraic counterpart of a CW-complex X is its chain complex C∗X. By definition, a chain complex C∗ is a sequence of abelian groups {Cn} together with homomorphisms dn : Cn → Cn−1 called the differ- entials such that dn dn+1 = 0. The condition dn dn+1 = 0 is equiv- alent to the requirement that the image of the homomorphism dn+1

belongs to the kernel of the homomorphism dn. Then, the so-called homology groups Hn(C∗) = ker dn/ im dn+1 can be defined.

The chain complex C∗X of a CW-complex X consists of the free abelian groups CnX over the generators [en

α ] corresponding to the n-cells of X. The boundary map dn : CnX → Cn−1X is defined by taking a generator [en

α ] into the linear combination ∑β kα,β[en−1 β ] of generators of Cn−1X

where kα,β is the number of times that the boundary of the cell en α

wraps around the cell en−1 β . More precisely, the coefficient kα,β is the

degree of the map

Sn−1 = ∂Dn α

α−→ Xn−1 → Xn−1/(Xn−1 \ en−1 β ) ≈ Sn−1.

The homology groups of the chain complex C∗X of a CW-complex X are simply denoted by HnX.

The dual of a chain complex (C∗, d∗) is a cochain complex (C∗, δ∗). It consists of abelian groups Cn = Hom(Cn, Z), and coboundary homo- morphisms δn : Cn → Cn+1. In other words, a cochain f in Cn is a linear function on chains in Cn. It is convenient to write f , x for the value f (x) of the function f on x. In this notation, the coboundary homomor- phism is defined by δn f , x = f , dn+1x. It follows that δn+1 δn = 0, and therefore the cohomology groups Hn(C∗) = ker δn/ im δn−1 can be defined. The cohomology group of the chain complex C∗X of a CW-complex X are denoted by HnX.

background information 13

A map of chain complexes f : C∗ → C′∗ is a family of maps of abelian groups fn : Cn → C′n such that d fn = fn−1 d. The requirement d fn = fn−1 d ensures that f gives rise to homomorphisms Hn(C∗) → Hn(C′∗) of homology groups and Hn(C∗) → Hn(C′∗) of cohomology groups.

A cellular map f : X → Y of CW-complexes is a map that takes n-cells of X to the n-th skeleton Yn. By the so-called Cellular Approxima- tion Theorem, every continuous map of CW-complexes can be contin- uously deformed to a cellular map.

A cellular map f defines a map C∗X → C∗Y of chain complexes by f ([en

α ]) = ∑ kα,β[en β] where kα,β is the number of times the disc f (en

α)

raps around the cell en β. More precisely, the coefficient kα,β is the degree

of the map

f−→ Yn/(Yn \ en β) ≈ Sn.

Dually, a cellular map f : X → Y of CW-complexes defines a map C∗Y → C∗X of cochain complexes.

Products.

When C and C′ are chain complexes, a new chain complex C ⊗ C′

can be defined; its n-th entry is ∑i+j=n Ci ⊗ C′j, while the boundary

homomorphism d(c⊗ c′) = dc⊗ c′+(−1)kc⊗ dc′ where k is the degree of c. For example, when X and Y are finite CW-complex, then X × Y is also a finite CW-complex that consists of products of cells in X and cells in Y. The chain complex C∗(X × Y) therefore is the tensor product C∗X ⊗ C∗Y of the chain complexes of X and Y. Similarly, the cochain complex C∗(X × Y) is the tensor product C∗X ⊗ C∗Y of cochain complexes.

Unfortunately, the inclusion of the diagonal X → X × X is not a cellular map. However, it can be approximated by a cellular map : X → X × X. It has the property that its composition with any of the two projections X × X → X is homotopic to the identity map. The cellular map defines the cup product

` : CnX⊗ CmX → Cn+mX

by identifying an element x⊗ y with an element in Cm+n(X × X) and then setting x ` y = ∗(x⊗ y). The cup product is a chain map and therefore defines a map on cohomology groups. The slant product

/ : Cn(X×Y)⊗ CkY → Cn−kX

14 elements of surgery theory

3 Hint: Let [1] denote a generator of H0X, and [X] a generator of H2X. Since the composition of with any of the two projections X × X → X is homo- topic to the identity map, we deduce that ∗[1] = [1]⊗ [1] and ∗[X] = [1]⊗ [X] + [X] ⊗ [1] + · · · , where · · · stands for terms which do not involve [1] and [X]. Note that every cell ai maps under into the diagonal in the torus whose meridian and parallel are copies of ai . Therefore ∗(ai) = [1] ⊗ ai + ai ⊗ [1]. Similarly, ∗(bi) = [1] ⊗ bi + bi ⊗ [1]. Since

αi ⊗ β j, ∗[X] = ∗(αi ⊗ β j), [X],

we deduce that

+∑[ai ]⊗ [bi ]− [bi ]⊗ [ai ].

[X] ∩ βi = ∗[X]/βi = ai .

is defined by first writing a chain a ∈ Cn(X × Y) as ∑ ai ⊗ a′i where ai ∈ C∗X and a′i ∈ C∗Y, and then setting a/b = ∑ b(a′i)ai. It is a chain map and therefore it defines a map on homology and cohomology groups. Together with the diagonal map , the slant product defines the cap product:

∩ : CnX⊗ CkX → Cn−kX

by a ∩ b = ∗a/b. Again, it is a chain map that induces a homomor- phism ∩ : HnX⊗ HkX → Hn−kX.

Example 0.6. The homology groups of a closed oriented surface X of genus g can be identified with Z in degrees 0 and 2, and with the group Za1, b1, ..., ag, bg in degree 1. Determine all product maps for X. 3

Figure 1.1: Lev Semenovich Pontryagin (1908–1988)

1

Stable homotopy groups of spheres

A quick look at the table of homotopy groups of spheres suffices to observe that the groups πm+kSk on the diagonals are the same for all sufficiently high values of k and any fixed m. For example, each of the groups πkSk is isomorphic to Z. These are the groups on the left- most non-zero diagonal in Table 1.1. The groups π1+kSk on the next diagonal to the right are isomorphic to Z2 for k > 2, while the groups π2+kSk are isomorphic to Z2 for k > 1. In fact, we will see that for any m the groups πm+kSk are the same at least for k > m + 1. These are so-called stable homotopy groups of spheres, denoted by πS

m. In today’s lecture we will recall the historically first approach to computing πS

m due to Lev Pontryagin. Later we will use the Pontryagin construction to prove the observed stabilization phenomenon, and compute the sta- ble groups πS

0 , πS 1 , πS

2 and πS 3 following the ideas of Pontryagin and

Rokhlin.

π1 π2 π3 π4 π5 π6 π7 π8 π9 π10 π11 S1 Z 0 0 0 0 0 0 0 0 0 0

S2 0 Z Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2

S3 0 0 Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2

S4 0 0 0 Z Z2 Z2 Z×Z12 Z2

2 Z2 2 Z24 ×Z3 Z15

S5 0 0 0 0 Z Z2 Z2 Z24 Z2 Z2 Z2

S6 0 0 0 0 0 Z Z2 Z2 Z24 0 Z

S7 0 0 0 0 0 0 Z Z2 Z2 Z24 0

Table 1.1: For every m, the groups πm+kSk on the m-th diagonal stabilize as k increases to infinity. The shaded ar- eas corresponds to the stable range k > m + 1.

It is worth mentioning that the Pontryagin approach admits an exten- sive generalization—the Pontryagin-Thom construction—which plays an indispensable role in contemporary homotopy theory.

To explain the Pontryagin construction we will need to review the notions of a submanifold in a Euclidean space, smooth function on a submanifold, tangent and normal vectors, frame, exponential map,

16 elements of surgery theory

Figure 1.3: As subsets of R2, the circle A is a closed submanifold, while the open interval D is an open submanifold. The curves B, C and E are not submanifolds for their end points, double point, and the corner respectively.

and tubular neighborhood of a manifold in a Euclidean space. The reader familiar with these notions may skip the following section.

1.1 Framed submanifolds in Euclidean spaces

In order to compute stable homotopy groups of spheres, Pontryagin came up with an idea of replacing stable homotopy groups of spheres with cobordism classes of framed smooth submanifolds in Euclidean spaces.

Informally, a smooth submanifold M of dimension m in Rm+k is a subset that locally looks like an open subset of Rm, see Figure 1.2. More precisely, a subset M ⊂ Rm+k is a smooth submanifold of di- mension m if for each point x in M there is a diffeomorphism Ψ of a neighborhood V of the origin in Rm+k into a neighborhood U of x ∈ Rm+k such that Ψ−1(M ∩U) is the intersection of V with the m- plane Rm ⊂ Rm+k defined in the standard coordinates (x1, ..., xm+k) by the equations xm+1 = · · · = xm+k = 0.

Figure 1.2: A submanifold M in Rm+k .

It is also common to say that M is a manifold placed in Rm+k, or, simply, a manifold in Rm+k. A closed submanifold in Rm+k is one whose underlying set is compact, while an open manifold is one with no closed components. You may see examples of submanifolds and non-submanifolds in Figure 1.3.

Manifolds in Rm+k are ubiquitous. For example, almost all fibers of any smooth map f : Rm+k → Rk are smooth (possibly empty) man- ifolds in Rm+k. Indeed, recall that a value y ∈ Rk is regular if the

stable homotopy groups of spheres 17

1 Sard Theorem. The set of non-regular values of a smooth map has measure zero.

2 Rank Theorem. Let f be a map from an open subset U of Rm+k to Rk such that the differential of f is surjective at a point x in U. Then there are coordinates (x1, ..., xm+k) in a neighborhood of x and coordinates in a neighborhood of f (x) such that f (x1, ..., xm+k) = (x1, ..., xk).

Figure 1.5: The tangent and the perpen- dicular spaces. Here, the tangent space is spanned by the vectors v1 = ∂ψ

∂x1 (0)

3 Hint to Exercise 1.1. The vectors ∂ψ ∂xi

(0) are the first m column vectors in the in- vertible Jacobi matrix of Ψ at x = 0. By the equation 1.1, the vectors ∂ψ

∂xi (0) span

Tx M.

differential of f at x is an epimorphism for every point x in the inverse image of y. By the Sard theorem,1 almost every point in the target is a regular value of f . On the other hand, by the Rank Theorem,2 the inverse image of a regular value is a manifold in Rm+k.

The restriction ψ = Ψ|V ∩Rm of the diffeomorphism Ψ in the defini- tion of a submanifold is called a coordinate chart on M. A coordinate chart ψ = ψ(x) locally parametrizes the submanifold M by means of m parameters x = (x1, ..., xm). In particular, for any function f on M, e.g., for f : M → Rn with n ≥ 0, the composition f ψ is a function in terms of (x1, ..., xm); it is called the coordinate representation of f over the coordinate chart ψ. A function f on M is smooth if for any coordinate chart ψ, the partial derivatives of all orders of the function f ψ are continuous.

Figure 1.4: A curve x(t) and its image ψ x(t).

Consider an arbitrary curve x = x(t) on V ∩Rm passing at the time t = 0 through the origin with velocity vector x(0) = (a1, ..., am). Then the corresponding curve ψ(x(t)) on M passes at the time t = 0 through x = ψ(0). Its velocity vector at t = 0 is

d(ψ x) dt

(0) = a1 ∂ψ

∂x1 (0) + a2

∂xm (0). (1.1)

Such a vector in Rm+k is called a tangent vector of M at x. The expres- sion (1.1) shows that the space Tx M of tangent vectors at x is a vector space spanned by the m vectors ∂ψ

∂xi (0) in Rm+k.

Exercise 1.1. Show that the vectors ∂ψ ∂xi

(0) are linearly independent and form a basis for the tangent space Tx M. Conclude that the dimension of the tangent space Tx M is the same as the dimension of M, i.e., m.3

A vector v in Rm+k at the point x of M is a perpendicular vector if it is orthogonal to the tangent space Tx M. The set of perpendicular vectors at x also forms a vector space, denoted by T⊥x M. We note that for any

18 elements of surgery theory

4 More precisely, a basis {τ1, ..., τk} for T⊥x M defines an isomorphism τx by τx(ei) = τi for i = 1, ..., k, where {ei} is the standard basis for Rk .

5 Tubular Neighborhood Theorem: If M is compact, then for small ε, the map exp is a diffeomorphism onto a neighborhood of M.

Figure 1.6: A tubular neighborhood of a framed manifold M consists of ε-discs centered at points x of M and orthogonal to Tx M.

point x ∈ M, there is a canonical isomorphism Tx M⊕ T⊥x M ≈ Rm+k

of vector spaces.

Choosing a basis for T⊥x M is equivalent to choosing an isomorphism τx : Rk → T⊥x M of vector spaces.4 Suppose that such an isomorphism τx is chosen for each point x. Then for each point x, the image τx(v) of any vector v ∈ Rk is a linear combination

τx(v) = α1(x)e1 + α2(x)e2 + · · ·+ αm+k(x)em+k

of basis vectors {e1, ..., em+k} of Rm+k. If the coefficient functions αi

are continuous (respectively, smooth) for every vector v, then we say that the manifold M in Rm+k is continuously (respectively, smoothly) framed. To summarize, a continuous (respectively, smooth) frame on a manifold M is a choice of a basis in each perpendicular vector space T⊥x M that changes continuously (respectively, smoothly) with x.

Up to continuous deformations, the set of continuous frames over M is isomorphic to the set of smooth frames over M. For this reason we will turn to smooth frames whenever it is necessary, and use continuous frames otherwise.

Given a (smoothly) framed manifold M with a frame τ, there is an exponential map exp : M × Dε → Rm+k where Dε is the open disc of radius ε in Rk centered at the origin. The exponential map is defined by exp(x, v) = x + τx(v). For example, when ε = ∞, the open disc Dε

coincides with Rk and the image of x × Dε is actually the perpendic- ular space T⊥x M ⊂ Rm+k. In general, perpendicular spaces T⊥x M and T⊥y M at different points x and y of M may have common points in Rm+k. However, if M is compact, then, by the Tubular Neighborhood Theorem,5 for small ε the exponential map is a diffeomorphism onto a neighborhood of M, called a tubular neighborhood of M. In other words, for small values of ε, the exponential map identifies a neighborhood of M in Rm+k with a tube M× Dε.

1.2 Cobordisms of framed manifolds

The Pontryagin construction shows that a class in πm+kSk defines a (closed) framed manifold M uniquely up to a cobordism. In fact, there is a bijective correpsondence between framed manifolds of dimension m in Rm+k up to cobordisms and pointed maps Sm+k → Sk up to ho- motopy, where cobordisms of m-manifolds in Rm+k is an equivalence relation which we define next.

stable homotopy groups of spheres 19

Figure 1.7: A manifold with bound- ary, and the collar neighborhood of its boundary.

6 Hint to Exercise 1.3. Let f : Rm+k → Rm+k be the parallel translation that brings M0 to M1. Let j : [0, 1] → [0, 1] be a smooth function that equals 0 for t < ε, increases on the interval (ε, 1− ε) and equals 1 for t > 1− ε. Then

Ft(x) = (1− j(t))x + j(t) f (x)

is a smooth deformation of M0 to M1 where t ∈ [0, 1] and x ∈ M0, while the trace W = {(Ft(x), t)} of the deforma- tion is a cobordism from M0 to M1. The frame over W at x is constructed in two steps. First, one applies dx(Ft) to the frame of M0 at x to get k vectors at Ft(x), and, second, one projects the k vectors to T⊥x W.

Definition 1.2. For a compact subset W in the slice Rm+k × [0, 1] of the Euclidean space Rm+k+1 and for i = 0, 1, let us denote the set of points of W on the hyperplane Rm+k × {i} by ∂iW. We say that the space W is a cobordism between ∂0W and ∂1W if the complement to ∂0W ∪ ∂1W in W is a manifold in Rm+k+1, and there is some ε > 0 such that

W ∩ (

= ∂1W × (1− ε, 1].

The union ∂W of the two sets ∂iW is called the boundary of W, while W is called a manifold with boundary. Note that the boundary ∂W has a nice neighborhood in W, namely the union of ∂0W × [0, ε) and ∂1W× (1− ε, 1], called a collar neighborhood. In particular, the boundary components ∂0W and ∂1W are smooth manifolds.

For example, for a manifold M in Rm+k the manifold with boundary W = M × [0, 1] in Rm+k × [0, 1] is a cobordism between two copies of M; it is called the trivial cobordism of M. On the other hand, W =

M× [0, 1/2) is not a cobordism between M and an empty set since W is not compact.

The technical requirement that the boundary of W possesses a col- lar neighborhood is often omitted. However, the existence of collar neighborhoods simplifies proofs. For example, in the presence of col- lar neighborhoods, the cobordism relation is clearly transitive.

Suppose now that the manifold W is framed, i.e., at each point (x, t) ∈ Rm+k× [0, 1] on the manifold W there is a chosen frame τ(x,t). Since the vectors in τ(x,t) are perpendicular to T(x,t)W, the frame τ(x,t) restricted to ∂0W and ∂1W turns the two boundary components into framed manifolds in Rm+k. Furthermore, suppose that τ(x,t) = τ(x,1) if t is ε- close to 1 and τ(x,t) = τ(x,0) if t is ε-close to 0 for some ε > 0. Then we say that W is a (framed) cobordism between the framed manifolds ∂0W and ∂1W.

The framed cobordism relation is an equivalence relation. The equiva- lence class of a framed manifold M will be denoted by [M].

Exercise 1.3. Show that a framed manifold M0 is cobordant to the framed manifold M1 obtained from M0 by a parallel translation.6

Finally, we note that the set of all cobordism classes of closed framed manifolds of dimension m in Rm+k forms an abelian group. Indeed, if M0 and M1 are two framed manifolds representing cobordism classes [M0] and [M1], we can use a parallel translation to shift M0 into the left

20 elements of surgery theory

7 If M0 and M1 do not share common points in Rm+k , then we may define the sum [M0] + [M1] to be [M0 ∪ M1] with- out taking parallel translations. How- ever, if the intersection of M0 and M1 is non-empty, the subset M0 ∪M1 of Rm+k

may not be a submanifold.

Figure 1.8: The inverse to the class of the framed manifold M is the class of the re- flection of M

8 To be more precise, we add collar neighborhoods to the boundary of W, and shrink Rm+k+1 along the (m + k + 1)-st direction so that W is a subset of Rm+k × [0, 1).

Figure 1.9: The Pontryagin construction can alternatively be described as follows: for each point x ∈ M consider its open ε-disc neighborhood Dx in the perpen- dicular space T⊥x M. These discs form a neighborhood U of M. To define the map f |U represent the sphere Sk as the the disc Dk in which all points of the boundary ∂Dk are identified. The map f sends Dx isomorphically onto Dk in such a way that the frame vectors τ1, ..., τk in Dx at x are send to the standard vectors e1, ..., ek on Dk at 0. Finally the map f is extended over Sm+k by sending the com- plement to U onto the south pole of Sk .

half space L = (−∞, 0)×Rm+k−1 of Rm+k and M1 into the right half space R = (0, ∞)×Rm+k−1, and get a new framed manifold M0 tM1

in Rm+k representing the sum [M0] + [M1].7 The so-defined operation (addition) is well-defined: changing representatives in the equivalence classes [M0] and [M1] does not change the resulting class [M0] + [M1].

The addition is clearly associative and commutative. The zero is rep- resented by an empty manifold. To construct the inverse of the cobor- dism class of a framed manifold M, place M into L and rotate L in Rm+k+1

+ = Rm+k × [0, ∞) along Rm+k−1 till L turns into R. Then the framed manifold M traces a framed cobordism W from M ⊂ L to a framed manifold −M ⊂ R.8 In particular, [M] + [−M] = 0, and there- fore −M represents the negative of [M], see Figure 1.8.

To summarize, we have shown that the cobordism classes of (closed) framed manifolds of dimension m in Rm+k form an abelian group.

1.3 The Pontryagin construction

Pontryagin observed a beautiful relation between framed manifolds M and homotopy groups of spheres. To see the relation, let us identify the complement in Sn to its south pole with Rn so that 0 ∈ Rn is the north pole of Sn; the south pole ∞ will be the base point of Sn. Let U denote an ε-tubular neighborhood of a closed framed manifold M of dimension m in Rm+k. Define f |U to be the map from U to Rk ⊂ Sk

by exp(x, v) 7→ v/(ε− |v|). It takes the manifold M to the north pole of Sk and identifies each fiber Dε of the tubular neighborhood with the Euclidean space Rk ⊂ Sk. Then, define the restriction f |Sm+k \U to be the constant map to ∞ ∈ Sk. The obtained continuous map f sends the south pole of Sm+k to the south pole of Sk, and, therefore, represents an element in πm+kSk. Choosing a different value for ε leads to the same element in πm+kSk. Thus, every closed framed manifold M in Rm+k determines a homotopy class in πm+kSk.

The corresondence [M] → [ f ] is well-defined as cobordant closed framed manifolds M0 and M1 define homotopic maps f0, f1 : Sm+k → Sk. Indeed, the Pontryagin construction applied to a cobordism W ⊂ Rm+k× [0, 1] between M0 and M1 results in a homotopy Sm+k× [0, 1]→ Sk between f0 and f1.

Conversely, let f : Sm+k → Sk be a smooth representative of an element in πm+kSk. In view of the Sard theorem, by slightly perturbing the

stable homotopy groups of spheres 21

9 Let us show that dx f |T⊥x M is an iso- morphism of the perpendicular space T⊥x M and the tangent space T0Rk . To begin with, the differential dx f is surjec- tive since the north pole of Sk is a reg- ular value of f . On the other hand, the map f takes the entire manifold M to 0, and therefore the differential dx f is triv- ial on Tx M. Consequently, the restric- tion dx f |T⊥x M is surjective. Finally, the dimension of the perpendicular space T⊥x M is the same as the dimension of T0Rk , which implies that the surjective homomorphism dx f |T⊥x M is actually an isomorphism.

10 We say that a map g is homotopic to h if there is a continuous family of maps ft parametrized by t ∈ [0, 1] such that f0 = g and f1 = h.

Figure 1.10: A framed manifold M ⊂ Rm+k also belongs to Rm+k+1. Its frame in Rm+k can be completed by em+k+1 to a frame in Rm+k+1.

map f , we may assume that the north pole of Sk is a regular value of f . Then the inverse image of the north pole is a manifold M in Rm+k ⊂ Sm+k, see Figure 1.9. Furthermore, at each point x in M the differential dx f takes the perpendicular space T⊥x M isomorphically onto the tangent space T0Rk = T0Sk at the origin. 9 Thus d f composed with the canonical isomorphism T0Rk = Rk defines a frame on M. In other words M is a framed manifold. However, the framed manifold M is not uniquely determined by the class in πm+kSk; choosing a different representative f of the class leads to a different framed manifold M.

To analyze the indeterminacy, let f0 and f1 be two different smooth representatives of the same element in πm+kSk determining two framed manifolds M0 and M1. Then f0 is homotopic10 to f1 through a smooth homotopy ft : Sm+k → Sk parametrized by t ∈ [0, 1]. Define a map f : Sm+k × [0, 1]→ Sk by f (x, t) = ft(x). We may assume that the north pole of Sk is a regular value of f0, f1 and f , and that the deformation f is trivial when the time parameter is close to 0 and 1. Then, as above, the map f determines a framed manifold W = f−1(0) in Rm+k × [0, 1] with boundary. Furthermore, the boundary of W coincides with the union of M0 and M1, and the frame of W restricts to the frames of M0

and M1. Thus W is a cobordism between the framed manifolds M0

and M1.

To summarize, we have shown that the homotopy classes in πm+kSk

are in bijective correspondence with the cobordism classes of framed manifolds.

Futhermore, the sum of cobordism classes of framed manifolds under the Pontryagin construction corresponds to addition in πm+kSk. Thus, we proved the Pontryagin Theorem.

Theorem 1.4 (Pontryagin). The group of cobordism classes of framed man- ifolds Mm ⊂ Rm+k is isomorphic to the homotopy group πm+kSk.

In the remainder of this section we will describe the stability phe- nomenon, and state the Pontryagin theorem in the form that we dis- cussed in the beginning of the lecture.

A framed manifold in Rm+k is naturally a framed manifold in a bigger space Rm+k+1; in the bigger space the frame at a point x ∈ M contains the old k frame vectors in Rm+k and the additional new basis vector em+k+1. Furthermore, any two cobordant framed manifolds in Rm+k

are also cobordant as framed manifolds in Rm+k+1. Consequently, in view of the Pontryagin theorem, there is a so-called Freudenthal homo-

22 elements of surgery theory

11 Recall that πS m denotes the m-th stable

homotopy group of spheres. It is iso- morphic to the group πm+kSk for any k > m + 1.

12 A frame τ1, ..., τk at a point p deter- mines a homomorphism of vector spaces

Rk −→ Rk

ei 7→ τi .

The group GLk(R) of homomorphisms has two path compoments; the compo- nent of homomorphisms f with det f > 0 and the component of homomor- phisms f with det f < 0. Therefore, each frame {τ1, ..., τk} can be smoothly deformed either to the basis {e1, ..., ek} or to the basis {−e1, ..., ek}.

Figure 1.11: A cobordism of framed points.

morphism πm+kSk → πm+k+1Sk+1. We will see later that it is an isomor- phism provided that k > m + 1 and it is an epimorphism for k > m, see the Freudenthal Theorem 2.4. Thus there is a characterization of stable homotopy groups of spheres in terms of framed manifolds.

Corollary 1.5 (Pontryagin). The group of cobordism classes of framed man- ifolds Mm ⊂ Rm+k is isomorphic to πS

m provided that k > m + 1.11

1.4 Example: The stable group πS 0

The Pontryagin construction identifies the stable homotopy group πS 0

with the cobordism group of (closed) framed manifolds of dimension 0 in Rk for k > 1. Being compact, such a manifold consists of finitely many points p, each of which equipped with a frame, i.e., a basis for TpRk ≈ Rk. As parallel translations do not change the cobordism class of a framed manifold (see Exercise 1.3), the exact location of points p is not essential.

A frame at any point p can be smoothly deformed12 into the standard positive basis {e1, ..., ek} or the standard negative basis {−e1, e2, ..., ek}. In particular, every class in πS

0 can be represented by a union of m positively framed points and n negatively framed points for some m, n ≥ 0. We claim that the homomorphism H : πS

0 → Z given by (m, n) 7→ m− n is well-defined and it is an isomoprhism.

Indeed, a cobordism W between two framed manifolds ∂0W and ∂1W of dimension 0 is a manifold with boundary of dimension 1, i.e., a union of finitely many circles and segments. We may discard all the circles from W, and still have a cobordism from ∂0W to ∂1W. Any remaining segment in W has two boundary components p and q. If both p and q belong to the same boundary component of W, i.e., either both belong to ∂0W or ∂1W, then the signs of their frames are opposite. On the other hand if p and q belong to the different components of W, then the signs of their frames are the same. This implies that H is well- defined as a map of sets. The disjoint union of two framed manifolds (m, n) and (m′, n′) is a framed manifold (m+m′, n+n′). The equalities

H(m + m′, n + n′) = (m + m′)− (n + n′) = H(m, n) + H(m′, n′)

show, then, that the map H is actually a homomorphism.

The homomorphism H is an epimorphism since every positive integer m is the image of the cobordism class of m positively framed points,

stable homotopy groups of spheres 23

13 The homomorphism H assigns, of course, to the homotopy class of a map Sk → Sk its degree.

while every negative integer n is the image of the cobordism class of n negatively framed points. Finally, the cobordism class of m positively framed points and m negatively framed points is trivial, which implies that the homomorphism H is injective. This completes the proof that H is an isomorphism of the stable homotopy group πS

0 onto Z. 13

1.5 Further reading

We only discussed manifolds of dimension m in Rm+k. These are usu- ally defined in advanced calculus courses, e.g., see Advanced calculus of several variables by C. H. Edwards, JR. In more advanced courses manifolds are defined without referring to the ambient space Rm+k, e.g., see the textbook Differential topology by M. W. Hirsch and An introduction to differentiable manifolds and Riemannian geometry by W. M. Boothby. Early works of L. S. Pontryagin on stable homotopy groups of spheres include Homotopy classification of the mappings of an (n+2)-dimensional sphere on an n-dimensional one (1950) and Smooth manifolds and their applications in homotopy theory (1955, 1976). Pontryagin applied his construction to compute πS

1 and πS 2 . The

Thom’s generalization of the Pontryagin construction can be found in his excellent paper Quelques propriétés globales des variétés dif- férentiables (1954), which lays the foundation of the cobordism the- ory. The J-homomorphism was defined by G. W. Whitehead in On the homotopy groups of spheres and rotation groups (1942).

The homotopy theoretic calculation of π3S2 as well as πS 1 can be found

in the textbook Topology and geometry by G. E. Bredon [Br93, page 465]. It is remarkable that even before the invention of higher ho- motopy groups by E. Cech in Höherdimensionale Homotopiegrup- pen [Ce32] and W. Hurewicz in Beiträge zur Topologie der Defor- mationen [Hu35], Heinz Hopf proved in Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche [Ho30] that there are infinitely many mutually non-homotopic pointed maps f : S3 → S2. Hopf used the linking number of links f−1x and f−1y for any two regular values x, y of f to distinguish the homotopy classes of maps f . Note that if x is the north pole of S2 and y a point near x, then f−1x and f−1y play the roles of the manifold M and its frame in the Pontryagin construction.

Figure 2.1: Steve Smale (b. 1930) https://math.berkeley.edu/ smale/

Figure 2.2: Immersion vs. non- immersion.

2

Immersion Theory

A manifold placed into Rn with possible self-intersection points is called an immersed manifold. The 50s and 60s saw a rapid develop- ment of the study of immersed manifolds. Historically, the birth of the Immersion Theory is commonly associated with the works of Stephen Smale who classified immersions of spheres, and, in particular, proved a peculiar, counterintuitive statement that the standard sphere S2 in R3 can be turned inside out by a deformation through immersions. The work of Smale was later extended by Hirsch to a classification of immersions of arbitrary manifolds.

In this lecture we will give an exposition of a modern approach to classical theorems of immersion theory, due to Rourke and Sander- son. In the prerequisite section §2.1 we introduce an ambient isotopy, which is a deformation of a manifold in Rm+k. In §2.2 we review the compression technique of Rourke and Sanderson. Smale-Hirsch the- orem is one of its applications (§99.2.2). Another application of the Rourke-Sanderson technique is the Freudenthal suspension theorem (§2.3) which we have discussed in chapter 1. We will conclude the lecture with two examples: the Smale paradox (§99.2.3), and the inter- pretation of stable homotopy groups in terms of cobordism groups of orientable immersed manifolds of dimension m in Rm+1 (§99.2.4).

To put the results of the present chapter into perspective, let’s recall that according to the Pontryagin theorem, every element of the sta- ble homotopy group πm+kSk is represented by a framed manifold M in Rm+k. We would like to simplify the representing manifold M as much as possible. In the present chapter we will use a deformation (ambient isotopy) to bring the manifold M into Rm+k−1 ⊂ Rm+k when possible. In later chapters we will attempt to replace M with a sphere

immersion theory 25

1 Let’s take a look at a simple exam- ple of how a smooth vector field turns into a differential equation. Suppose that w(x, y) is a vector field on R2

with components (x2 + y, y3), while γ(t) is a curve on R2 with components (x(t), y(t)). Then the velocity vector of γ(t) has components (x(t), y(t)), and the differential equation γ(t) = w(γ(t)) corresponding to the vector field w(x, y) takes the form{

x = x2 + y y = y3

where x = x(t) and y = y(t) are func- tions in t.

2 Existence and Uniqueness Theorem for first order differential equations. Suppose that the function F is continu- ous. Then the differential equation y(t) = F(t, y(t)) with an initial value y(t0) = y0 has a unique solution on the interval [t0 − ε, t0 + ε] for some real number ε < 0.

Figure 2.3: A vector field w and the trace γ(t) of isotopy.

3 In other words, a smooth vector field is a smooth function

w : R×Rm+k → Rm+k ,

w : (t, x) 7→ wt(x).

We may think of wt as of a smooth fam- ily of vector fields parametrized by t.

by performing spherical surgeries on M.

2.1 Ambient isotopy

In this section we will make precise the somewhat vague notion of a deformation of a manifold in Rm+k. The correct term is actually an ambient isotopy. When possible, we will use an ambient isotopy to simplify a framed manifold M in Rm+k by deforming it into the Euclidean hyperspace Rm+k−1.

A smooth vector field w on Rm+k defines a first order differential equa- tion γ(t) = w(γ(t)) with smooth coefficients;1 its solution is a curve γ : R → Rm+k whose velocity vector at the moment t is precisely the vector w at the point γ(t) ∈ Rm+k. Informally, if we imagine that the space Rm+k consists of particles, then the vector field w shows the direction for the flow of the particles, see Fig. 2.3.

We will often assume that the vectors in the vector field w are bounded in length, i.e., there is a positive number l such that |w(x)| < l for all x ∈ M. By the Existence and Uniqueness Theorem2, if the vectors w(x) of the vector field w are bounded in length, then there is a unique solution γ(t) = F(x, t) for any initial condition x = γ(0). In other words, a vector field defines a one parametric deformation F of the Euclidean space Rm+k:

Ft : Rm+k → Rm+k,

Ft : x 7→ F(x, t),

with F0 being the identity map of Rm+k. It is known that the deforma- tion (or, flow) F(x, t) associated with a smooth vector field is smooth in x and t. In fact, for each moment of time t, the map Ft is a a dif- feomorphism of Rm+k, i.e., a smooth homeomorphism whose inverse is also smooth.

Note that if we do not require that the vectors w(x) in the vector field w are bounded in length, then it may happen that the flow carries a point x = γ(0) to infinity in finite time, and therefore, the position γ(t) for large t may not be well-defined.

A smooth time dependent vector field wt smoothly3 associates at each moment of time t a vector wt(x) to each x ∈ Rm+k. Again, under the assumption that all vectors wt(x) are bounded in length, there exists a

26 elements of surgery theory

4 Let v0 be a normal vector field over M, and v1 be the vector field obtained from v by projecting each vector v0(x) to the perpendicular space T⊥x M. The lin- ear deformation of v0 to v1 is the fam- ily of vector fields vt = (1 − t)v0 + tv1 parametrized by t ∈ [0, 1].

5 Hint for Exercise 2.1. Note that the dif- ferential dFt of the diffeomorphism Ft is invertible. In particular, if a vector v(x) is not in Tx M, then the vector dFt(v(x)) is not in Ty Mt, where Mt = Ft(M) and y = Ft(x). This implies that normal vec- tor fields over M flow into normal vector fields over Mt. On the other hand, the differential dFt may not preserve the an- gles, i.e., perpendicular vector fields over M may not flow into perpendicular vec- tor fields over Mt.

flow Ft : Rm+k → Rm+k along wt in the sense that for each x the curve t 7→ Ft(x) has velocity wt(Ft(x)).

The flow F of a (possibly, time dependent) vector field is an (ambient) isotopy, i.e., it is a smooth parametric family Ft of diffeomorphisms of Rm+k such that F0 is the identity diffeomorphism. In practice, an ambient isotopy is viewed as a parametric deformation of M in Rm+k, as for each moment of time t, the set Mt = Ft(M) is also a manifold placed in Rm+k. We even say that Ft is an ambient isotopy of M. However, as the word "ambient" suggests, the flow F actually deforms not only M, but the entire background space Rm+k.

A smooth (respectively, continuous) normal vector field v on a manifold M in Rm+k is a function that associates to each point x in M a vector at x not in Tx M such that the components of the vector v(x) change smoothly (respectively, continuously) with the parameter x. Project- ing each vector v(x) to the corresponding perpendicular plane T⊥x M produces a vector field that is actually perpendicular to Tx M. Every perpendicular vector field is normal. Not ever normal vector field is perpendicular, but every normal vector field can be linearly deformed to a perpendicular one.4 Exercise 2.1 shows why we prefer to work not only with perpendicular vector fields.

Exercise 2.1. Under isotopy a manifold with a normal vector field flows into a manifold with a normal vector field. On the other hand, a manifold with a perpendicular vector field may not flow into a mani- fold with a perpendicular vector field. 5

Exercise 2.2. Let M be a manifold of dimension m in Rm+k. Suppose that normal vector fields v0 and v1 over M are homotopic through normal vector fields. Show that v0 is isotopic to v1, i.e., there is an ambient isotopy Ft with t ∈ [0, 1] such that F0 = id, Ft(x) = x for all x in M, and dF1(v0) = v1.

2.2 The Global Compression Theorem

We say that a vector field v along a manifold M in Rm+k is vertical up if for every point x ∈ M the vector v(x) is a positive multiple of the last basis vector em+k; we regard Rm+k−1 × {0} ⊂ Rm+k as horizontal.

Compression Theorem 2.1 (Rourke-Sanderson). Every normal vector field v on a closed manifold M of dimension m in Rm+k with k > 1 can

immersion theory 27

Figure 2.4: A normal frame over M = S1

in R2 can not be vertical up since at the right most point, i.e., at the point (x, y) in M with maximal value of y, the verti- cal up direction is tangent to M. In R3

every vector v(x) can be rotated to the vertical up position.

Figure 2.5: A rotation of v to w.

Figure 2.6: The neighborhood U of M consists of δ-discs perpendicular to M and centered at points x in M. Over each of the discs we extend w so that it changes linearly along radial lines from w(x) at the center x of the disc to em+k at the boundary of the disc.

be made vertical up by an ambient isotopy of the manifold M and a smooth deformation of the vector field v through normal vector fields.

The condition k > 1 in the Compression theorem is important. For example, the (radial) unit perpendicular vector field v over the stan- dard circle M = S1 in R2 can not be straightened up by an ambient isotopy of M and a deformation of v through normal vector fields. We recommend that the reader proves this fact and finds a straighten- ing up deformation of v in R3 ⊃ R2 before reading the proof of the Compression theorem, see Figure 2.4.

On the other hand, in view of Exercise 2.2 we may prohibit deforma- tions of vector fields in the Compression Theorem: Under the hypoth- esis of the Compression Theorem, there is an ambient isotopy Ft with t ∈ [0, 1] such that F0 is the identity map of Rm+k and dF1(v) = em+k

over F1(M).

Proof. To begin with we deform v into the vector field of unit length perpendicular to M. Then each vector v(x) determines a point on the unit sphere Sm+k−1; namely, if we translate v(x) so that its initial point is the origin in Rm+k−1, then its terminal point specifies a point on the unit sphere Sm+k−1. In other words, the vector field v defines a so-called Gauss map M → Sm+k−1. By the Sard Theorem, almost every value in the target of the Gauss map is regular. On the other hand, if k > 1, then regular values are precisely those with empty inverse image. Thus, if k > 1, then by slightly rotating the manifold M together with the vector field, we may assume that the image of the Gauss map avoids the south pole. That is to say, the vector field v is never vertical down.

Next we observe that since M is compact, the angle between v(x) and the vertical direction −em+k is at least ε > 0. Choose a positive real number µ < ε, and rotate each vector v(x) in the plane P spanned by v(x) and ek+m in the direction vertically up till its last component is positive and v(x) has angle at least µ with the horizontal plane. Such a rotation of the vector field v can be chosen to be through normal vector fields over M. Indeed, the intersection of Tx M with the plane of rotation P is either a point or a line whose angle with the horizontal plane is at least ε; therefore we stop rotating v(x) before reaching Tx M. Let w denote the smoothing of the rotated unit vector field over M.

The vector field w can be extended over Rk+m so that the angle of w with the horizontal plane is at least µ and outside a δ-neighborhood

28 elements of surgery theory

6 Note that since the manifold M is com- pact, it lies in a ball of finite radius. On the other hand, since the angle of w with the horizontal hyperspace is at least µ, the flow along w lifts each point by at least T sin µ units at time T.

7 Hint for Exercise 2.3. To define such an isotopy, let f be a function on π(M) that assigns to a point π(x) the last co- ordinate xm+k of the point x ∈ M. As a smooth function on M, the function f admits an extension to a smooth func- tion on all Rm+k−1. Now, define a vector field w on Rm+k by w(x, y) = − f (x)em+k where (x, y) ∈ Rm+k−1 ×R. Show that the flow of the vector field w is along the direction ±em+k and brings M to π(M) in time 1.

Figure 2.7: A framed manifold M ⊂ Rm+k also belongs to Rm+k+1. Its frame in Rm+k can be augmented by em+k+1 to form a frame in Rm+k+1.

U of M the vector field w is vertical up, for some small δ. Indeed, the neighborhood U consists of δ-discs perpendicular to M and centered at points x in M. Over each of the discs we extend w so that it changes linearly along radial lines from w(x) at the center x of the disc to em+k

at the boundary of the disc. Finally, we extend w over the rest of Rm+k

by vertical up vector filed, smooth the resulting vector field w, and multiply it by a bump function with (a very big) compact support. The flow of w isotopes M to a manifold M′ outside U in finite time.6

The modified normal vector field on M′ is vertical up.

The Compression Theorem allows us in certain cases to deform M to the hyperplane Rm+k−1. Indeed, suppose that a compact manifold M is equipped with a vertical up normal vector field v. Furthermore, suppose that none of the lines in the direction v intersects M at more than one point. Then the projection π : Rm+k → Rm+k−1 along em+k

places M into the hyperplane Rm+k−1.

Exercise 2.3. Show that there is an ambient isotopy that moves all points of Rm+k in the vertical direction and brings M to π(M). 7

We say that the ambient isotopy of Exercise 2.3 compresses the manifold M to Rm+k−1.

The Compression Theorem admits various generalizations, which we discuss in the chapter of additional topics. For example, the Multi- compression theorem asserts that if a manifold M of dimension m in Rm+k is equipped with n < k linearly independent perpendicular vec- tor fields v1, ..., vn, then there is an ambient isotopy Ft that straightens the vectors v1, ..., vn up in the sense that F0 = id and dF1(vi) = ei for i = 1, ..., n.

2.3 The Freudenthal suspension theorem

Recall that the Pontryagin construction identifies the homotopy groups πm+kSk of spheres with the cobordism classes of framed manifolds M of dimension m in the Euclidean space Rm+k. Such a manifold M also lies in a bigger space Rm+k+1. Furthermore, its frame in Rm+k can be augmented with an additional vector em+k+1 to produce a frame in Rm+k+1, see Figure 2.7. In other words, each framed manifold in Rm+k may also be considered to be a framed manifold in a bigger space Rm+k+1. Furthermore, cobordant framed manifolds in Rm+k are

immersion theory 29

8 Freudenthal homomorphism (classical definition): Every map f : Sm+k → Sk

defines the suspension map Sm+k+1 → Sk+1 by taking the poles of Sm+k+1 into the respective poles of Sk+1 and taking the longitude through any point x in Sm+k homeomorphically into the longi- tude through the point f (x). Taking suspensions of representatives, gives an- other definition of the Freudenthal ho- momorphism πm+kSk → πm+1Sk+1.

9 To show surjectivity of the Freuden- thal homomorphism, we need to show that the framed manifold M ⊂ Rm+k+1

is cobordant to a manifold in Rm+k ⊂ Rm+k+1 with v being vertically up. Re- call also that isotopies of framed mani- folds as well as deformations of normal vector fields imply cobordisms.

Figure 2.8: The manifold of distinct pairs (x, y) is obtained from the manifold M× M by removing the diagonal {x = y}.

10 Note that the proof of injectivity of the Freudenthal homomorphism does not follow immediately from the proof of surjectivity. We begin with a framed manifold M in Rm+k that is zero cobor- dant in Rm+k+1 by means of a framed cobordism W. Of course, we may find a flow F that straightens up the last vector v of the framed cobordism W, and then we can compress W to Rm+k . However if F displaces M, the resulting compressed cobordism W is a cobordism to zero of a displaced manifold M, not the original manifold M.

still cobordant as framed manifolds in Rm+k+1. In view of the Pon- tryagin construction, the correspondence between framed manifolds in Rm+k and Rm+k+1 defines a so-called Freudenthal suspension homo- morphism8 of homotopy groups of spheres πm+kSk → πm+k+1Sk+1.

Theorem 2.4. The Freudenthal homomorphism is an isomorphism for k >

m + 1 and an epimorphism for k > m.

It follows that the homotopy groups of spheres πm+kSk are the same for a fixed m and k > m + 1. These are stable homotopy groups πS

m.

Proof of the Freudenthal theorem. Suppose that k > m. Recall that a nor- mal frame v1, .., vk+1 on a manifold M in Rm+k+1 is a set of k + 1 normal vector fields over M that project to a basis of T⊥x M for each point x ∈ M. In view of the Compression theorem, we may assume that the (k + 1)-st normal vector field v = vk+1 is vertical up.9

We claim that the manifold M can be rotated slightly in Rm+k+1 so that any line in the direction em+k+1 intersects M at most at one point. Indeed, for any two distinct points x and y in M, the unit vector w(x, y) in the direction x− y points to a point in Sm+k. Since the dimension of pairs of distinct points is 2m, and the dimension of the sphere Sm+k is at least 2m + 1, the set of vectors w(x, y) is of measure zero by the Sard theorem, see Figure 2.8. Therefore by slightly rotating M we may make sure that em+k−1 is not among the vectors w(x, y), which precisely means that any line in the direction em+k+1 intersects M at most at one point. Of course, after the rotation v may not be vertically up any more. However, since we may choose the rotation to be arbitrarily small, the rotated v can be deformed back to the vertical up position.

Now we may use the vector field v to compress the framed manifold M ⊂ Rm+k+1 to a framed manifold in Rm+k, see Exercise 2.3. The compression can be iterated consecutively using the vectors vk+1, vk, ... as long as the index of the vector vi under consideration satisfies i > m. This proves surjectivity of the Freudenthal homomorphism.

To show that the Freudenthal homomorphism is injective,10 suppose that a framed manifold M ⊂ Rm+k is zero cobordant after applying the Freudenthal suspension. In other words, after including M into Rm+k+1 and adding the vector field em+k+1 to its frame, the mani- fold M becomes zero cobordant, i.e., there is a framed manifold W in Rm+k+1 × [0, 1] with boundary M ⊂ Rm+k+1 × {0}. We know that the last vector field v in the frame of W restricts to em+k+1 over M. In fact,

30 elements of surgery theory

11 We can extend v vertically up over the slice Rm+k+1 × [0, ε) since, by the defi- nition of a framed cobordism, the only points of W that are in Rm+k+1 × [0, ε) are the points of the collar neighborhood U.

Figure 2.9: The framed manifold M is placed in Rm+k which is depicted as the direction to the right. It is zero cobor- dant in the horizontal space Rm+k+1. The cobordism W itself is a manifold with boundary in Rm+k+1 × [0, 1]. If we can compress W to Rm+k × [0, 1] without displacing M, then we get a cobordism to zero of the original framed manifold M in Rm+k .

we may assume that it restricts to em+k+1 over an ε-collar neighbor- hood U of M in W.

Now let us carefully examine the argument in the proof of the Com- pression Theorem. We first deform v over W to a vector field so that the last component of each vector v(x) is positive. We may assume that during the deformation v stays vertically up over the collar neigh- borhood U, see Figure 2.9. Next we extend v to a vector field on Rm+k+1 × [0, 1]. We can extend v as in the Compression Theorem so that v is vertically up outside a neighborhood of W. In addition we may assume that v is vertically up over the slices Rm+k+1 × [0, ε) and Rm+k+1× (1− ε, 1] whose union we will denote by V.11 Then the flow F along the extended vector field is well-defined as the condition that v ≡ em+k+1 over V prevents the points of Rm+k+1 × [0, 1] from flow- ing out of the region of definition of F. The flow F takes W outside its neighborhood in a finite time T and therefore straightens up the vector field v. Note that all points in U travel the distance T in the direction v = em+k+1. Therefore, if we postcompose the ambient isotopy along v with translation along −Tem+k+1, then we get a framed cobordism W ′

bounding the original manifold M and such that v is vertical up.

Finally, when k > m + 1, we can use the same argument as in the proof of surjectivity to show that W ′ compresses to a cobordism of M in Rm+k × [0, 1]. Thus, a framed manifold M ⊂ Rm+k is cobordant to zero after the Freudenthal suspension only if it is cobordant to zero itself.

2.4 Further reading

We borrowed the proof of the Compression Theorem from the original paper The compression theorem I [RS01] of C. Rourke and B. Sander- son. The theorem has many interesting applications some of which are explained in The compression theorem II: directed embeddings [RS01a] and The compression theorem III: applications [RS03]. We gave one more application here: the Freudenthal theorem.

The original proof of Freudenthal suspension theorem in Über die Klassen der Sphärenabbildungen. I. Große Dimensionen [Fr38] was deemed hard. The modern homotopy theoretic proof is based on the Blackers-Massey excision theorem proved in The homotopy groups of a triad [BM51]. Recall that in general the Freudenthal homomorphism E is not an isomorphism below the stable range. In the paper On the

immersion theory 31

12 To find the EHP exact sequence, Whitehead observed that E is the ho- momorphism induced by an embedding Sn → Sn+1, and therefore E fits the ho- motopy long exact sequence of the pair (Sn+1, Sn).

Freudenthal theorem [Wh53], Whitehead fitted E into the so-called EHP sequence12

· · · → πqSn E−→ πq+1Sn+1 H−→ πq−1S2n−1 P−→ πq−1Sn → · · · .

Promptly James proved in On the iterated suspension [J54] the ex- istence of a similar long exact sequence for an iterated Freudenthal homomorphism Em : πqSn → πq+mSn+m. Its geometric interpretation in terms of Pontryagin construction can be found in the paper Geo- metric interpretations of the generalized Hopf invariant [KS77] by Koschorke and Sanderson.

The original proof of the Smale theorem appeared in his papers A classification of immersions of the two-sphere and The classifica- tion of immersions of spheres in Euclidean spaces, [Sm58, Sm59]. Smale’s theorems were generalized by Hirsch in the paper Immer- sions of manifolds[Hi59]. It was observed later that not only immer- sions can be replaced with their formal counterparts. A number of remarkable discoveries culminated in the so-called homotopy princi- ple (h-principle, for short). Besides the classical reference of Partial differential relations [Gr86] by Gromov, we recommend the Introduc- tion to the h-principle [EM02] by Eliashberg and Mishachev.

Figure 3.1: Marston Morse (1892–1977)

3

Spherical surgeries

Under the Pontryagin construction, every element of the homotopy group πm+kSk is identified with the cobordism class of framed man- ifolds of dimension m in Rm+k. The choice of a representing framed manifold in the cobordism class is far from being unique, and it is our goal to find a simple representative. Our strategy is to begin with an arbitrary (a priory complicated) representative and then use a framed cobordism to simplify it as much as possible.

A general cobordism on a manifold W0 could be overly complex: it is hard to describe such a cobordism and its action on the homotopy groups πiW0. However, we will see that every non-trivial cobordism breaks into a composition of elementary, so-called spherical cobordisms. A spherical cobordism on a manifold W0 performs a spherical surgery on W0. A spherical surgery is relatively easy to describe in terms of (locally flat) topological submanifolds of Rm+k which are almost every- where smooth. We will discuss a general Cairns-Whitehead technique of smoothing topological manifolds, and apply it to smooth corners and more complicated singularities produced by an embedded spher- ical surgery.

Next, we will learn how to efficiently encode a spherical surgery in terms of a base of surgery. We will determine how a spherical surgery on a manifold W0 changes its homotopy groups. As a demonstration of the introduced surgery technique, we will calculate the homotopy group π3S2 as well as the stable homotopy groups πS

1 and πS 2 .

spherical surgeries 33

1 Recall, the differential dx f of the height function linearly projects the space TxW ⊂ Rm+k ×R to the last com- ponent.

Figure 3.2: A cobordism W between W0 and W1 with two critical points. The tan- gent planes at critical points are horizon- tal.

Figure 3.3: After slightly perturbing W from Figure 3.2, the height function has only Morse critical points.

3.1 Surgery and cobordisms

Trivial cobordisms

Recall that a cobordism of framed manifolds of dimension m in Rm+k

is a framed closed manifold W of dimension m+ 1 in Rm+k× [0, 1]. The projection f of W onto the last coordinate is a so-called hight function. We say that a point x ∈ W is regular if dx f is surjective, and critical otherwise.1 Geometrically, the critical points of a height function are those at which the tangent plane is horizontal, see Figure 3.2.

There is a simple coordinate description of regular and critical points. Namely, by the Rank theorem, in a neighborhood of a regular point, there are coordinates (x1, ..., xm, xm+1) on W such that f (x) = xm+1. On the other hand, after slightly perturbing W, near each critical point there are coordinates such that

f (x1, ..., xm+1) = −x2 1 − · · · − x2

i + x2 i+1 + · · · x2

m+1,

see Figure 3.3. The latter statement—which we will not prove here—is known as the Morse lemma, while the perturbed function is called a Morse function.

The integer i in the coordinate representation of f near a critical point p may be different for different critical points. It is called the index of the critical point. The index of a critical point p does not depend on the choice of a coordinate chart about p. Note that if p is a critical point of f of index i, then p is also a critical point of − f of index j = m + 1− i.

For example, in the Figure below the cobordism has two Morse critical points, the points p1 and p2. At the point p1 the height function f has a local minimum; in a neighborhood of p1 in appropriate coordinates f can be written as f (x1, x2) = x2

1 + x2 2. The index of the critical point p1

is 0. The point p2 is a saddle point. In its neighborhood in appropriate coordinates the height function has the form f (x1, x2) = −x2

1 + x2 2. In

particular, the index of p2 is 1.

According to Lemma 3.1, as t changes from 0 to 1, the regular levels Wt = f−1(t), i.e., levels with no critical points of f , change by isotopy.

34 elements of surgery theory

Figure 3.4: The vertically up vector field em+k+1 and its projection v to W.

2 Hint to Exercise 3.2 We have defined the map

: W0 × [0, 1]→W

in the proof of Lemma 3.1 by (x, t) = Ft(x). The flow F carries points along the scaled vector field w. Since the last component of the scaled vector field w is 1, the last component of each point in- creases with velocity 1. Thus, in time t, the map Ft lifts each point of W0 to the hight t. In other words, (W0 × {t}) ⊂ Wt. Since the flow F−1 along the nega- tive of the scaled vector field w is smooth and brings Wt to W0 we deduce that |W0 × {t} is a diffeomorphism. In par- ticular, identifies W0 × {t} with Wt. Since is a homeomorphism and d is of full rank, it follows that is a diffeo- morphism, e.g., see [Lee13].

Lemma 3.1. Suppose that f has no critical points. Then the cobordism W is trivial, that is there is a diffeomorphism : W0 × [0, 1] → W that identifies W0 × {t} with the t-th level Wt of f for all t.

Sketch of the proof. Consider the vector field em+k+1 in the slice Rm+k× [0, 1] of a Euclidean space. Since the hight function f has no critical points, over W the projection v of em+k+1 to W is never horizontal, i.e., the last component vm+k+1 of the vector field v is never zero. If we scalar multiply v by the smooth function 1/vm+k+1, then we obtain a vector field w over W with last component 1. The flow F of the manifold W0 along the scaled vector field w carries W0 along W and brings it to Wt at the time t. In fact, F defines a desired diffeomorphism by taking a point (x, t) to Ft(x).

Note that in the proof of Lemma 3.1 we need to scale the vector field v in order to make sure that for any point x ∈ ∂0W the last coordinate of the curve t 7→ Ft(x) increases with speed 1 so that the flow Ft indeed brings W0 to the t-th level at the time t.

Exercise 3.2. Show that the map is a diffeomorphism that identifies W0 × {t} with Wt. In particular, show that −1 is smooth.2

Spherical cobordisms

In view of Lemma 3.1 as t increases from 0 to 1 the level Wt essen- tially changes only when t passes a critical value, i.e, the value f (x) of a critical point x. In appropriate coordinates, the gradient of f in a neighborhood of a Morse critical point is

d f (x1, ..., xm+1) = 2(−x1, ...,−xi, xi+1, ..., xm+1),

spherical surgeries 35

j ,

(−2x1, ...,−2xi , 2xi+1 + 2xm+1),

or, in other words, d f (x) = 0 only if x = 0.

Figure 3.5: By perturbing W, we may as- sume that each level Wt contains at most one critical point of the hight function.

4 Hint for Exercise 3.3. Suppose that the hight function on the cobordism W has n critical points p1, ..., pn. We may stretch Rm+k × [0, 1] along the (m + k + 1)-st co- ordinate in such a way that for the new hight function f on W we have f (pi) ∈ (i − 1, i). Next we may modify W near Wi by ambient isotopy so that each level Wt near Wi for i = 1, ..., n− 1 is obtained from Wi by a vertical translation. Then W is a composition of spherical cobor- disms, and it is diffeomorphic to the original cobordism.

5 Near the critical point of index i, in Morse coordinate neighborhood, the core disc of a spherical surgery consists of the points (x1, ..., xi , 0, ..., 0), while the belt disc consists of the points (0, ..., 0, xi+1, ..., xm+1).

which means that the set of critical points of a Morse function is dis- crete.3 It is even finite, since W is compact. Furthermore, by slightly perturbing W, we may always assume that each level Wt contains at most one critical point of f , see Figure 3.5.

Thus, every non-trivial cobordism is a composition of spherical cobor- disms W whose height function has at most one critical point. Under a spherical cobordism, the manifold ∂0W = W0 is modified into a mani- fold ∂1W = W1 by the so-called spherical surgery.

Exercise 3.3. Let W be a non-trivial cobordism such that each level Wt

contains at most one critical point of its hight function. Show that W is diffeomorphic to a composition of spherical cobordisms.4

Let’s describe the result of a spherical surgery. The projection v of the vertical up vector field over W to W is a smooth vector field. It is zero only at the critical point p of the height function since only at the critical point p the tangent space TpW is horizontal. Let Fv denote the flow along the projected vector field v. The set of points x in W with Fv

t (x) → p as t → ∞ is called the core disc of the spherical surgery, see Figure 3.6. Similarly, the set of points x with Fv

t (x) → p as t → −∞ is called the belt disc of the spherical surgery.5 We will see shortly that the core and belt discs are indeed discs Di of dimension i and Dj of dimension j = m = 1− i respectively. The boundary ∂Di is called the attaching sphere of the surgery, while ∂Dj is called the belt sphere. The attaching sphere Si−1 is a sphere on the level W0, while the belt sphere Sj−1 is one on the level W1.

All points on W0 \ Si−1 are carried by the flow Fv t (x) to points on W1 \

Sj−1. The flow of the vector field −v defines the inverse map showing that the manifold W0 \ Si−1 is diffeomorphic to W1 \ Sj−1. In fact, we may choose neighborhoods hi and hj of the core and belt spheres so that the flow defines a diffeomorphism between W0 \ hi and W1 \ hj. Thus, we determined a geometric description of a spherical surgery: up to isotopy the manifold W1 is obtained from W0 by replacing hi

with hj, see Figure 3.7.

To identify hi and hj, we may assume that the unique critical point p of the hight function is on the level W1/2. Since f has no other critical points, we already know that all levels Wt for t ∈ [0, 1/2− ε]

are mutually diffeomorphic. The same is true for levels Wt with t ∈ [1/2+ ε, 1]. In fact, we may remove f−1[0, 1/2− ε) and f−1(1/2+ ε, 1] from the cobordism W without changing its diffeomorphism type. To simplify notation, we will denote the parts of the core and the cocore

36 elements of surgery theory

Figure 3.6: The belt disc Dj is above the critical point (red), the core disc Di is be- low the critical point (red). The traces of the flow F outside the belt and core disc (blue) run from W0 to W1.

Figure 3.7: The handles hi and hj are in green. The flow F carries W0 \ hi to W1 \ hj.

Figure 3.8: The manifolds W1/2−ε and W1/2+ε.

discs Di and Dj remaining in W by the same symbols, and call their boundaries the attaching and belt spheres.

The manifolds W1/2+ε and W1/2−ε in a neighborhood of the critical point p are depicted in Figure 3.8. The integral curves of the flow Ft

are perpendicular to the level sets Wt, see Figure 3.9. In particular, the core disc Di is a disc in the plane (x1, ..., xi, 0, ..., 0), while the cocore disc Dj is a disc in the plane (0, ..., 0, xi+1, ..., xm+1), see Figure 3.8. The neighborhood hi of the attaching sphere Si−1 in W1/2−ε is diffeomor- phic to Si−1×Dj. The rest of W1/2−ε flows under Ft to the complement in W1/2+ε to a neighborhood hj of the belt sphere Sj−1. The neighbor- hood hj is diffeomorphic to Di × Sj−1, and W1/2+ε is obtained from W1/2−ε by removing hi, transferring the remaining part by Ft to the level 1/2 + ε, and, finally, attaching hj.

Example 3.4. There is a spherical surgery of index 1 on a sphere S2

that results in a torus, see Figure 3.10. A spherical surgery of index 1 removes a neighborhood h1 = S0 × D2 of an attaching sphere S0, and then attaches a handle h2 = D1 × S1, which is a cylinder. If we turn the corresponding cobordism W upside down, then we obtain a cobordism corresponding to a spherical surgery turning a torus into a sphere S2, see Figure 3.11. Now the spherical surgery is of index 2. It removes a neighborhood h2 = S1 × D2 of an attaching sphere S1, and then attaches a handle h1 = D2 × S0.

Conversely, let W0 be a manifold of dimension m in Rm+k, and let Di

be a disc of dimension i in Rm+k together with j linearly independent perpendicular vector fields v1, ..., vj over Di such that ∂Di is a sphere in W0 while all vector fields v`|∂Di are tangent to W0. Suppose that Di

intersects the manifold W0 only along ∂Di. Then, we say that the disc Di together with the j perpendicular vector fields is a base of spherical surgery. We call Di the core disc.

In view of the exponential map, the disc Di together with the perpen- dicular vector fields v` are often replaced with a so-called embedded i-handle Hi ⊂ Rm+k which is a thickening Di × Dj of the core disc Di

in Rm+k, see Figure 3.12. In fact, given a base of surgery (Di, v1, ..., vj), we may choose an embedding ψ : Di × Dj → Rm+k of the i-handle Hi so that ψ(Di × {0}) is the core disc, the i-handle Hi intersects W0

along ∂Di × Dj, and ψ({0} × ei) = εvi for some small real number ε

and every basis vector ei in the standard unit disc Dj ⊂ Rj.

It can be shown that in the presence of a base of spherical surgery, there exists a spherical surgery of W0 to a manifold W1 such that hi

spherical surgeries 37

Figure 3.9: The curves of the flow Ft.

Figure 3.10: A surgery turning a 2- sphere into a torus.

Figure 3.11: A surgery turning a torus into a 2-sphere.

is given by ∂Hi ∩W0, while W1 is obtained from W0 \ hi by taking its union with hj = ∂Hi \ hi and smoothing the corners. To construct the corresponding spherical cobordism W, we begin with a collar neigh- borhood W0 × [0, 1/2] in Rm+k × [0, 1/2], and attach to it a copy of Hi

in Rm+k × {1/2}, see Figure 3.13. The interior of the obtained space is a manifold with corners. Its boundary is a copy of a manifold with corners W1. Finally, we attach to the constructed space the final part W1 × [1/2, 1] in Rm+k × [1/2, 1]. The resulting cobordism

W = (W0 × [0, 1/2]) ∪ Hi ∪ (W1 × [1/2, 1])

is a so-called topological manifold. We can smooth it, and turn W into a smooth cobordism between smooth manifolds W0 and W1.

Example 3.5. The spherical surgery in Figure 3.10 corresponds to the base of surgery in Figure 3.14. Note that the base of surgery of index n consists of a disc Dn of dimension n together with dim W0 − dim ∂Dn

perpendicular vector fields along Dn. The base of surgery of index 1 in Figure 3.14 consists of the disc of dimension 1 as well as 2 perpendicu- lar vector fields. In Figure 3.15, the surgery is of index 2. Therefore the base of surgery consists of a disc of dimension 2 as well as 1 normal vector field.

3.2 Framed base of surgery

In the previous section we have seen how a base of surgery defines a spherical surgery. However, we are still to answer the question, "Given a base of surgery on a framed manifold, does it correspond to a framed surgery?" Recall that we are interested in framed surgeries rather than surgeries as the Pontryagin construction identifies homotopies of maps of spheres with framed cobordisms rather than cobordisms.

We will see that the answer to the question is negative in general: in order to define a framed surgery, we need a base of framed surgery rather than a base of surgery.

We will say that Rm+k = Rm+k×{0} is the horizontal space in Rm+k+1

while the last factor in Rm+k+1 will be referred to as the vertical space. Let W0 be a manifold of dimension m in the horizontal space Rm+k × {0} framed by vector fields τ1, ..., τk. Let Di be a disc in Rm+k ×

E L E M E N T S O F S U R G E R Y T H E O R Y

Contents

1 Stable homotopy groups of spheres 15

1.1 Framed submanifolds in Euclidean spaces 16

1.2 Cobordisms of framed manifolds 18

1.3 The Pontryagin construction 20

1.4 Example: The stable group πS 0 22

1.5 Further reading 23

2 Immersion Theory 24

2.1 Ambient isotopy 25

2.4 Further reading 30

3 Spherical surgeries 32

3.2 Framed base of surgery 37

3.3 Reduction of homotopy groups 38

3.4 Examples: Stable homotopy groups πS 1 and πS

2 39

4.1 The Whitney weak embedding and immersion theorems 45

4.2 The Whitney trick 46

4.3 Strong Whitney embedding theorem 50

4.4 Strong Whitney immersion theorem 51

4.5 Solution to Exercises 53

4.6 Further reading 53

5.1 The Wall representative and the invariant µ 56

5.2 Homotopy spheres 61

5.4 Main theorems 66

5.5 Further reading 67

6.1 Invariance of signature and Kervaire invariant 71

6.2 The main theorems for even dimensional manifolds 73

6.3 The main theorems for odd dimensional manifolds 76

6.4 Solutions to exercises 79

7 Characteristic classes of vector bundles 84

7.1 Vector bundles over topological spaces 84

7.2 Characteristic classes of vector bundles 85

7.2.1 The Euler class 86

7.2.2 Stiefel-Whitney classes 87

7.2.3 Chern classes 89

7.2.4 Pontryagin classes 89

8.1 Maps of SU(2) and Spin(4) to orthogonal groups 94

8.1.1 The map SU(2)→ SO(3) 94

8.1.2 The map Spin(4)→ SO(4) 94

8.2 Milnor’s fiber bundles 95

4

8.4 Invariants of Milnor’s fibrations 98

8.5 Exotic spheres of dimension 7 100

8.6 Further reading 100

9 Exotic spheres 102

9.1 The group of smooth structures on a sphere 102

9.2 The surgery long exact sequence for spheres 105

9.3 Calculation of the group Pm 106

9.4 Calculation of the group Nm. 108

9.5 Calculation of homomorphisms in the surgery exact sequence 110

9.6 Hirzebruch genera and computation of the L-genus 111

9.6.1 Multiplicative series 111

9.6.2 Genera of manifolds M4n with trivial p1, ..., pn−1. 113

9.7 Example: The third stable homotopy group πs 3 114

9.8 Further reading 115

10.1.1 Handle rearrangement 117

10.1.2 Handle creation 118

10.1.3 Handle cancellation 119

10.1.4 Handle slides 119

10.1.5 Handle trading 120

11 Surgery on maps of simply connected manifolds 127

11.1 Normal maps 127

11.2.1 Tensor products 130

11.2.3 Poincaré Duality 132

12 The surgery long exact sequence 137

5

13.1 The Whitney trick in non-simply connected manifolds 142

13.2 Right and left Modules 144

13.3 Homology and cohomology with twisted coefficients 145

13.4 Products in (co)homology with twisted coefficients 147

13.5 Poincaré complexes 148

13.6 Homological algebra 150

14.2 Surgery on maps of manifolds of even dimension 154

14.2.1 Wall representatives 154

14.2.2 The intersection pairing λ and self-intersection form µ 156

14.2.3 Effect of surgery on homotopy groups 159

14.2.4 The main theorem 159

14.3 Surgery on maps of manifolds of odd dimension 160

14.3.1 Heegaard splittings 160

14.3.2 ε-quadratic formations 163

14.4 Further reading 168

99 Additional Topics 171

99.1.1 Hopf fibration 172

99.1.3 The Pontryagin-Thom construction 174

99.1.4 The stable J-homomorphism 175

99.2 Topics for Chapter 2 177

99.2.1 The Multicompression theorem 177

99.2.2 The Smale-Hirsch Theorem 179

99.2.3 The Smale Paradox 181

99.2.4 Cobordisms of oriented immersed manifolds 182

99.2.5 The Local Compression Theorem 183

99.2.6 Solutions to Exercises 186

99.3 Topics for Chapter 3 189

99.3.1 Example: Cobordisms 189

6

99.4.1 Bilinear forms on free modules 194

99.5 Topics for Chapter 6 195

99.5.1 Heegaard splittings 195

99.5.2 Alternative proof of the surgery theorem for framed manifolds of odd dimension 196

99.6 Topics for Chapter 14 199

99.6.1 The Whitehead torsion 199

99.6.2 The s-cobordism theorem 200

Figure 1: A CW-complex is constructed by induction beginning with a (red) dis- crete set X0, by attaching (blue) seg- ments, then (grey) 2-dimensional cells and so on.

0

0.1 Homotopy groups

Pointed topological spaces. A pointed topological space is a topological space X together with a point ∗ ∈ X called the basepoint of X. A map of pointed topological spaces is required to take the basepoint in the domain to the basepoint in the target. The pointed union X ∨ Y of a pointed topological spaces X and Y is the space X tY/∼ in which the basepoints of X and Y are identified.

Example 0.1. Let Sn denote the standard unit sphere centered at ori- gin in the Euclidean space Rn+1 with basis e1, ..., en. The sphere Sn be- comes pointed if we pick its south pole, i.e., the end point of −en+1 to be the basepoint of Sn. The intersection of Sn with the hyperplane per- pendicular to e1 is a meridian Sn−1 of Sn. The quotient space Sn/Sn−1

is homeomorphic to the bouquet of spheres Sn ∨ Sn.

A homotopy of a map f : X → Y to a map g : X → Y is a continuous family of maps Ft : X → Y parametrized by t ∈ [0, 1] such that F0 = f and F1 = g. When the spaces X and Y are pointed, every map Ft in the homotopy is required to be pointed as well. A space X is homotopy equivalent to a space Y if there are maps f : X → Y and g : Y → X such that g f is homotopic to the identity map of X, and f g is homotopy equivalent to the identity map of Y.

CW complexes. Most topological spaces that we will encounter are CW complexes. A CW complex X is constructed inductively beginning with a discrete set X0 called the 0-skeleton of X. The n-skeleton Xn

is obtained from the (n − 1)-skeleton Xn−1 by attaching a collection

8 elements of surgery theory

tDn α of closed discs of dimension n with respect to attaching maps

α : ∂Dn α → Xn−1. In other words, Xn consists of Xn−1 and the discs

Dn α with each x in the boundary ∂Dn

α identified with α(x) ∈ Xn−1. We say that the interior of ∂Dn

α is an open cell en α . The so-constructed

space is a CW-complex X = ∪Xn. A set U in X is declared to be open (respectively, closed) if each intersection U ∩ Xn is open (respectively, closed).

Homotopy groups. For n ≥ 0, the set of homotopy classes [ f ] of pointed maps f : Sn → X into a pointed topological space is denoted by πnX. The set πnX is a group when n > 0 with the sum [ f ] + [g] represented by the composition

Sn −→ Sn/Sn−1 ≈ Sn ∨ Sn f∨g−−→ X,

where the first map is one collapsing the meridian of Sn into a point. The set π0X is the set of path components of X. The group π1X is the fundamental group. The homotopy groups πnX with n > 1 are abelian. A space X is n-connected if the groups πiX are trivial in the range i ≤ n. Thus, a non-empty space is −1-connected, a path connected space is 0-connected, while a simply connected space is 1-connected. A map f : X → Y of topological spaces induces a homomorphism f∗ : πiX → πiY of homotopy groups by [g] 7→ [ f g].

Theorem 0.2 (Whitehead). Two path-connected CW-complexes X and Y are homotopy equivalent if and only if there exists a map f : X → Y which induces an isomorphism of homotopy groups f∗ : πiX → πiY for all i ≥ 0.

Relative homotopy groups. A pair (X, A) of pointed topological spaces consists of a pointed topological space X and a pointed subspace A such that the distinguished point of A is the same as the distinguished point of X. A map of pairs (X, A) → (Y, B) is a pointed map from X to Y which takes the subspace A to B. Similarly, a homotopy of maps f0, f1 of pairs is a continuous family Ft of maps of pairs parametrized by t ∈ [0, 1] such that F0 = f0 and F1 = f1. Let Dn denote the stan- dard disc of dimension n, and Sn−1 its boundary. Then, the set of homotopy classes of maps from the pair (Dn, Sn−1) to a pair (X, A) is denoted by πn(X, A). It is a group if n ≥ 2, and an abelian group for n ≥ 3. We note that when the subspace A is a point, the relative ho- motopy groups reduce to homotopy groups: πn(X, ∗) = πnX. Every map of pairs f : (X, A) → (Y, B) induces a map f∗ of relative homo- topy groups π∗(X, A)→ π∗(Y, B). For any pair (X, A), there is a long exact sequence of homotopy groups:

· · · → πn+1(X, A) ∂−→ πn A i∗−→ πnX

j∗−→ πn(X, A)→ · · · ,

background information 9

where i : A → X is the inclusion, j is the inclusion (X, ∗) ⊂ (X, A)

of pairs, while ∂ takes the class [ f ] of a map f : Dn → X to the class [ f |∂Dn] of the map f |∂Dn : Sn−1 → A.

A pair (X, A) is said to be n-connected if πi(X, A) = 0 for i ≤ n and any choice of a base point ∗ ∈ A.

Theorem 0.3. Let X be a CW complex decomposed as the union of sub- complexes A and B with nonempty connected intersection C = A ∪ B. If (A, C) is m-connected and (B, C) is n-connected, m, n ≥ 0, then the map πi(A, C)→ πi(X, B) induced by inclusion is an isomorphism for i < m + n and a surjection for i = m + n.

Theorem 0.4 (Hatcher, Proposition 4.28). If a CW pair (X, A) is r-connected and A is s-connected, with r, s ≥ 0, then the map πi(X, A) → πi(X/A)

induced by the quotient map X → X/A is an isomorphism for i ≤ r + s and a surjection for i = r + s + 1.

Let A and B be two vector spaces over a field k, or finitely gener- ated free abelian groups (in which case we put define k to be the ring Z). We say that a bilinear function ψ : A ⊗ B → k is non-degenerate with respect to A if for any linear function f on A there is a unique vector b ∈ B such that f (a) = ψ(a, b). Similarly, we say that ψ is non- degenerate with respect to the second factor if for any function f on B there is a vector a ∈ A such that f (b) = ψ(a, b).

Theorem 0.5. Suppose that W is an oriented compact manifold of dimension 2q + 1 ≥ 5. Suppose that W and ∂W are q− 1 connected manifolds. Then the intersection pairing

πqW/Tor⊗ πq+1(W, ∂W)/Tor → Z

is non-degenerate with respect to the first factor, while the pairing

πq+1W/Tor⊗ πq(W, ∂W)/Tor → Z

is non-degenerate with respect to the second factor.

Proof. Let us prove the non-degeneracy of the first pairing. For the second pairing, the proof is similar.

By the Hurewicz isomorphism, the free part of πqW is isomorphic to the free part of HqW. Since the homology pairing is non-degenerate, for every function f on the free part of HqW there is an element x f in the free part of Hq+1(W, ∂W) such that f (y) = x f · y for all elements y

10 elements of surgery theory

1 See Exercise 23 in section 4.2 of Hatcher’s book.

in the free part of Hq+1(W, ∂W). By the Hurewicz theorem, the homo- morphism πq+1(W/∂W) → Hq+1(W/∂W) is surjective.1 Therefore, the element x f lifts to an element in πq+1(W/∂W). Finally, x f admits a lift with respect to the map πq+1(W, ∂W) → Hq+1(W/∂W) since such a map is surjective by Theorem 0.4.

For the second pairing, we have πq(W, ∂W) ≈ πq(W/∂W) is isomor- phic to Hq(W, ∂W). Therefore, for any function f on the part of this group, there is a corresponding element x f in Hq+1(W) which can be lifted to an element in πq+1W.

0.2 Bordisms and Cobordisms

Bordisms. We say that two maps f0 : M0 → X and f1 : M1 → X of closed oriented manifolds into a topological space are bordant, if there is a map f : W → X of a compact manifold W with boundary ∂W = M0 t (−M1) such that f |M0 = f0 and f |M1 = f1. The set of bordism classes of maps of manifolds of dimension m into a manifold N forms a group mX with operation given by taking the disjoint union of maps.

More generally, given a compact oriented manifold M, a continuous map f : (M, ∂M) → (X, A) is bordant to zero if there is a compact ori- ented manifold W with boundary ∂W = MtM′ and a map F : W → X such that F|M = f and F(M′) ⊂ A. Two maps fi : (Mi, ∂Mi)→ (X, A)

are said to be bordant if f0 t f1 : (M0 ∪−M1, ∂M0 ∪−∂M1)→ (X, A) is bordant to zero. The set of bordant maps of manifolds of dimension m forms an abelian group m(X, A). Clearly, we have m(X, ∅) = mX.

Every map g : (X′, A′) → (X, A) of pairs induces a homomorphism g∗ : ∗(X′, A′) → ∗(X, A) by [ f ] 7→ [g f ]. Homotopic maps g, g′

induce the same homomorphism g∗ = g′∗. There is a boundary ho- momorphism ∂ : m(X, A) → m−1 A defined by associating the class of f : (M, ∂M) → (X, A) with the class f |∂M : ∂M → A. For any pair (X, A) of spaces, there is a long exact sequence of abelian groups

→ m+1(X, A) ∂−→ m A→ mX → m(X, A)→,

where i : A → X and j : (X, ∅) → (X, A) are the inclusions. The re- duced bordism group mX is the kernel of the homomorphism ε∗ in- duced by the projection ε of X onto a one point space. There is an iso- morphism m(X, A) ≈ m(X/A) where X/∅ is the disjoint union of

background information 11

2 Note that oriented bordism classes are represented by maps of oriented closed manifolds, while oriented cobor- dism classes are represented by oriented proper maps of manifolds which are not necessarily oriented or closed.

X and a one point set. Suppose that U is a subset in X such that its clo- sure is in the interior of A. Then the inclusion (X \U, A \U)→ (X, A)

induces an isomorphism of bordism groups.

Cobordisms. We say that a proper map f : M→ N of manifolds is ori- ented if the normal bundle of a map f × i : M→ N×Rk for an embed- ding i : M → Rk with k 1 is oriented. Two orientations defined by i and i′ are equivalent, if they are compatible with an isotopy from i to i′. We say that two proper oriented maps f0 : M0 → N and f1 : M0 → N are cobordant if there is a proper oriented map f : W → N of a manifold with boundary ∂W = M0 ∪M1 extending f0 and f1 in such a way that the orientation of f agrees with the orientations of f0 and f1. The set of cobordism classes of maps of dimension −q = dim M− dim N into N is a group denoted by qN. 2

Every smooth map g : N′ → N of manifolds defines a homomorphism g∗ : ∗N → ∗N′ by [ f ] 7→ [g∗ f ]. The homomorphism g∗ depend only on the homotopy class of g. A proper oriented map g : N′ → N of dimension d defines the so-called Gysin homomorphism g! : mN′ → m−dN by [ f ] 7→ [g f ].

Products. The external products

∧ : mN ⊗m′N ′ → m+m′(N × N′),

∧ : dN ⊗d′N ′ → d+d′(N × N′)

are defined by [ f1] ∧ [ f2] = [ f1 × f2]. There are also homomorphisms

/ : q(N × N′)⊗mN′ → q−m(N),

\ : qN ⊗m(N × N′)→ m−q(N′).

The first homomorphism is defined by [ f ]/[g] = [(π1 f f ∗(1N × g)]. For example, suppose that both f and g are transverse embeddings. Then f f ∗(1N × g) is the inclusion into N × N′ of the intersection of the images of f and g. The map π1 projects it to N. The second homomorphism is defined by [ f ]\[g] = [π2 g g∗( f × 1N′)].

The group ∗N is a ring with operation given by [ f1] ∪ [ f2] = ∗( f1 ∧ f2) where : N → N × N is the diagonal map. There is also the ∩- product

∩ : qN ⊗m(N, ∂N) −→ m−q(N, ∂N),

defined by the composition

\−→ m−q(N, ∂N).

12 elements of surgery theory

When N is a compact orientable manifold of dimension n, the class [N] of the identity map of (N, ∂N) is called the fundamental class of N in n(N, ∂N).

0.3 Homology and Cohomology

Let X be a CW-complex. We recall that it is constructed by induction beginning with a discrete set X0. The n-th skeleton Xn is obtained from the (n− 1)-skeleton by attaching to Xn−1 closed discs Dn

α of dimension n by means of the attaching maps α : ∂Dn

α → Xn−1. The interior en α of

the attached disc Dn α is called an open cell.

The algebraic counterpart of a CW-complex X is its chain complex C∗X. By definition, a chain complex C∗ is a sequence of abelian groups {Cn} together with homomorphisms dn : Cn → Cn−1 called the differ- entials such that dn dn+1 = 0. The condition dn dn+1 = 0 is equiv- alent to the requirement that the image of the homomorphism dn+1

belongs to the kernel of the homomorphism dn. Then, the so-called homology groups Hn(C∗) = ker dn/ im dn+1 can be defined.

The chain complex C∗X of a CW-complex X consists of the free abelian groups CnX over the generators [en

α ] corresponding to the n-cells of X. The boundary map dn : CnX → Cn−1X is defined by taking a generator [en

α ] into the linear combination ∑β kα,β[en−1 β ] of generators of Cn−1X

where kα,β is the number of times that the boundary of the cell en α

wraps around the cell en−1 β . More precisely, the coefficient kα,β is the

degree of the map

Sn−1 = ∂Dn α

α−→ Xn−1 → Xn−1/(Xn−1 \ en−1 β ) ≈ Sn−1.

The homology groups of the chain complex C∗X of a CW-complex X are simply denoted by HnX.

The dual of a chain complex (C∗, d∗) is a cochain complex (C∗, δ∗). It consists of abelian groups Cn = Hom(Cn, Z), and coboundary homo- morphisms δn : Cn → Cn+1. In other words, a cochain f in Cn is a linear function on chains in Cn. It is convenient to write f , x for the value f (x) of the function f on x. In this notation, the coboundary homomor- phism is defined by δn f , x = f , dn+1x. It follows that δn+1 δn = 0, and therefore the cohomology groups Hn(C∗) = ker δn/ im δn−1 can be defined. The cohomology group of the chain complex C∗X of a CW-complex X are denoted by HnX.

background information 13

A map of chain complexes f : C∗ → C′∗ is a family of maps of abelian groups fn : Cn → C′n such that d fn = fn−1 d. The requirement d fn = fn−1 d ensures that f gives rise to homomorphisms Hn(C∗) → Hn(C′∗) of homology groups and Hn(C∗) → Hn(C′∗) of cohomology groups.

A cellular map f : X → Y of CW-complexes is a map that takes n-cells of X to the n-th skeleton Yn. By the so-called Cellular Approxima- tion Theorem, every continuous map of CW-complexes can be contin- uously deformed to a cellular map.

A cellular map f defines a map C∗X → C∗Y of chain complexes by f ([en

α ]) = ∑ kα,β[en β] where kα,β is the number of times the disc f (en

α)

raps around the cell en β. More precisely, the coefficient kα,β is the degree

of the map

f−→ Yn/(Yn \ en β) ≈ Sn.

Dually, a cellular map f : X → Y of CW-complexes defines a map C∗Y → C∗X of cochain complexes.

Products.

When C and C′ are chain complexes, a new chain complex C ⊗ C′

can be defined; its n-th entry is ∑i+j=n Ci ⊗ C′j, while the boundary

homomorphism d(c⊗ c′) = dc⊗ c′+(−1)kc⊗ dc′ where k is the degree of c. For example, when X and Y are finite CW-complex, then X × Y is also a finite CW-complex that consists of products of cells in X and cells in Y. The chain complex C∗(X × Y) therefore is the tensor product C∗X ⊗ C∗Y of the chain complexes of X and Y. Similarly, the cochain complex C∗(X × Y) is the tensor product C∗X ⊗ C∗Y of cochain complexes.

Unfortunately, the inclusion of the diagonal X → X × X is not a cellular map. However, it can be approximated by a cellular map : X → X × X. It has the property that its composition with any of the two projections X × X → X is homotopic to the identity map. The cellular map defines the cup product

` : CnX⊗ CmX → Cn+mX

by identifying an element x⊗ y with an element in Cm+n(X × X) and then setting x ` y = ∗(x⊗ y). The cup product is a chain map and therefore defines a map on cohomology groups. The slant product

/ : Cn(X×Y)⊗ CkY → Cn−kX

14 elements of surgery theory

3 Hint: Let [1] denote a generator of H0X, and [X] a generator of H2X. Since the composition of with any of the two projections X × X → X is homo- topic to the identity map, we deduce that ∗[1] = [1]⊗ [1] and ∗[X] = [1]⊗ [X] + [X] ⊗ [1] + · · · , where · · · stands for terms which do not involve [1] and [X]. Note that every cell ai maps under into the diagonal in the torus whose meridian and parallel are copies of ai . Therefore ∗(ai) = [1] ⊗ ai + ai ⊗ [1]. Similarly, ∗(bi) = [1] ⊗ bi + bi ⊗ [1]. Since

αi ⊗ β j, ∗[X] = ∗(αi ⊗ β j), [X],

we deduce that

+∑[ai ]⊗ [bi ]− [bi ]⊗ [ai ].

[X] ∩ βi = ∗[X]/βi = ai .

is defined by first writing a chain a ∈ Cn(X × Y) as ∑ ai ⊗ a′i where ai ∈ C∗X and a′i ∈ C∗Y, and then setting a/b = ∑ b(a′i)ai. It is a chain map and therefore it defines a map on homology and cohomology groups. Together with the diagonal map , the slant product defines the cap product:

∩ : CnX⊗ CkX → Cn−kX

by a ∩ b = ∗a/b. Again, it is a chain map that induces a homomor- phism ∩ : HnX⊗ HkX → Hn−kX.

Example 0.6. The homology groups of a closed oriented surface X of genus g can be identified with Z in degrees 0 and 2, and with the group Za1, b1, ..., ag, bg in degree 1. Determine all product maps for X. 3

Figure 1.1: Lev Semenovich Pontryagin (1908–1988)

1

Stable homotopy groups of spheres

A quick look at the table of homotopy groups of spheres suffices to observe that the groups πm+kSk on the diagonals are the same for all sufficiently high values of k and any fixed m. For example, each of the groups πkSk is isomorphic to Z. These are the groups on the left- most non-zero diagonal in Table 1.1. The groups π1+kSk on the next diagonal to the right are isomorphic to Z2 for k > 2, while the groups π2+kSk are isomorphic to Z2 for k > 1. In fact, we will see that for any m the groups πm+kSk are the same at least for k > m + 1. These are so-called stable homotopy groups of spheres, denoted by πS

m. In today’s lecture we will recall the historically first approach to computing πS

m due to Lev Pontryagin. Later we will use the Pontryagin construction to prove the observed stabilization phenomenon, and compute the sta- ble groups πS

0 , πS 1 , πS

2 and πS 3 following the ideas of Pontryagin and

Rokhlin.

π1 π2 π3 π4 π5 π6 π7 π8 π9 π10 π11 S1 Z 0 0 0 0 0 0 0 0 0 0

S2 0 Z Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2

S3 0 0 Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2

S4 0 0 0 Z Z2 Z2 Z×Z12 Z2

2 Z2 2 Z24 ×Z3 Z15

S5 0 0 0 0 Z Z2 Z2 Z24 Z2 Z2 Z2

S6 0 0 0 0 0 Z Z2 Z2 Z24 0 Z

S7 0 0 0 0 0 0 Z Z2 Z2 Z24 0

Table 1.1: For every m, the groups πm+kSk on the m-th diagonal stabilize as k increases to infinity. The shaded ar- eas corresponds to the stable range k > m + 1.

It is worth mentioning that the Pontryagin approach admits an exten- sive generalization—the Pontryagin-Thom construction—which plays an indispensable role in contemporary homotopy theory.

To explain the Pontryagin construction we will need to review the notions of a submanifold in a Euclidean space, smooth function on a submanifold, tangent and normal vectors, frame, exponential map,

16 elements of surgery theory

Figure 1.3: As subsets of R2, the circle A is a closed submanifold, while the open interval D is an open submanifold. The curves B, C and E are not submanifolds for their end points, double point, and the corner respectively.

and tubular neighborhood of a manifold in a Euclidean space. The reader familiar with these notions may skip the following section.

1.1 Framed submanifolds in Euclidean spaces

In order to compute stable homotopy groups of spheres, Pontryagin came up with an idea of replacing stable homotopy groups of spheres with cobordism classes of framed smooth submanifolds in Euclidean spaces.

Informally, a smooth submanifold M of dimension m in Rm+k is a subset that locally looks like an open subset of Rm, see Figure 1.2. More precisely, a subset M ⊂ Rm+k is a smooth submanifold of di- mension m if for each point x in M there is a diffeomorphism Ψ of a neighborhood V of the origin in Rm+k into a neighborhood U of x ∈ Rm+k such that Ψ−1(M ∩U) is the intersection of V with the m- plane Rm ⊂ Rm+k defined in the standard coordinates (x1, ..., xm+k) by the equations xm+1 = · · · = xm+k = 0.

Figure 1.2: A submanifold M in Rm+k .

It is also common to say that M is a manifold placed in Rm+k, or, simply, a manifold in Rm+k. A closed submanifold in Rm+k is one whose underlying set is compact, while an open manifold is one with no closed components. You may see examples of submanifolds and non-submanifolds in Figure 1.3.

Manifolds in Rm+k are ubiquitous. For example, almost all fibers of any smooth map f : Rm+k → Rk are smooth (possibly empty) man- ifolds in Rm+k. Indeed, recall that a value y ∈ Rk is regular if the

stable homotopy groups of spheres 17

1 Sard Theorem. The set of non-regular values of a smooth map has measure zero.

2 Rank Theorem. Let f be a map from an open subset U of Rm+k to Rk such that the differential of f is surjective at a point x in U. Then there are coordinates (x1, ..., xm+k) in a neighborhood of x and coordinates in a neighborhood of f (x) such that f (x1, ..., xm+k) = (x1, ..., xk).

Figure 1.5: The tangent and the perpen- dicular spaces. Here, the tangent space is spanned by the vectors v1 = ∂ψ

∂x1 (0)

3 Hint to Exercise 1.1. The vectors ∂ψ ∂xi

(0) are the first m column vectors in the in- vertible Jacobi matrix of Ψ at x = 0. By the equation 1.1, the vectors ∂ψ

∂xi (0) span

Tx M.

differential of f at x is an epimorphism for every point x in the inverse image of y. By the Sard theorem,1 almost every point in the target is a regular value of f . On the other hand, by the Rank Theorem,2 the inverse image of a regular value is a manifold in Rm+k.

The restriction ψ = Ψ|V ∩Rm of the diffeomorphism Ψ in the defini- tion of a submanifold is called a coordinate chart on M. A coordinate chart ψ = ψ(x) locally parametrizes the submanifold M by means of m parameters x = (x1, ..., xm). In particular, for any function f on M, e.g., for f : M → Rn with n ≥ 0, the composition f ψ is a function in terms of (x1, ..., xm); it is called the coordinate representation of f over the coordinate chart ψ. A function f on M is smooth if for any coordinate chart ψ, the partial derivatives of all orders of the function f ψ are continuous.

Figure 1.4: A curve x(t) and its image ψ x(t).

Consider an arbitrary curve x = x(t) on V ∩Rm passing at the time t = 0 through the origin with velocity vector x(0) = (a1, ..., am). Then the corresponding curve ψ(x(t)) on M passes at the time t = 0 through x = ψ(0). Its velocity vector at t = 0 is

d(ψ x) dt

(0) = a1 ∂ψ

∂x1 (0) + a2

∂xm (0). (1.1)

Such a vector in Rm+k is called a tangent vector of M at x. The expres- sion (1.1) shows that the space Tx M of tangent vectors at x is a vector space spanned by the m vectors ∂ψ

∂xi (0) in Rm+k.

Exercise 1.1. Show that the vectors ∂ψ ∂xi

(0) are linearly independent and form a basis for the tangent space Tx M. Conclude that the dimension of the tangent space Tx M is the same as the dimension of M, i.e., m.3

A vector v in Rm+k at the point x of M is a perpendicular vector if it is orthogonal to the tangent space Tx M. The set of perpendicular vectors at x also forms a vector space, denoted by T⊥x M. We note that for any

18 elements of surgery theory

4 More precisely, a basis {τ1, ..., τk} for T⊥x M defines an isomorphism τx by τx(ei) = τi for i = 1, ..., k, where {ei} is the standard basis for Rk .

5 Tubular Neighborhood Theorem: If M is compact, then for small ε, the map exp is a diffeomorphism onto a neighborhood of M.

Figure 1.6: A tubular neighborhood of a framed manifold M consists of ε-discs centered at points x of M and orthogonal to Tx M.

point x ∈ M, there is a canonical isomorphism Tx M⊕ T⊥x M ≈ Rm+k

of vector spaces.

Choosing a basis for T⊥x M is equivalent to choosing an isomorphism τx : Rk → T⊥x M of vector spaces.4 Suppose that such an isomorphism τx is chosen for each point x. Then for each point x, the image τx(v) of any vector v ∈ Rk is a linear combination

τx(v) = α1(x)e1 + α2(x)e2 + · · ·+ αm+k(x)em+k

of basis vectors {e1, ..., em+k} of Rm+k. If the coefficient functions αi

are continuous (respectively, smooth) for every vector v, then we say that the manifold M in Rm+k is continuously (respectively, smoothly) framed. To summarize, a continuous (respectively, smooth) frame on a manifold M is a choice of a basis in each perpendicular vector space T⊥x M that changes continuously (respectively, smoothly) with x.

Up to continuous deformations, the set of continuous frames over M is isomorphic to the set of smooth frames over M. For this reason we will turn to smooth frames whenever it is necessary, and use continuous frames otherwise.

Given a (smoothly) framed manifold M with a frame τ, there is an exponential map exp : M × Dε → Rm+k where Dε is the open disc of radius ε in Rk centered at the origin. The exponential map is defined by exp(x, v) = x + τx(v). For example, when ε = ∞, the open disc Dε

coincides with Rk and the image of x × Dε is actually the perpendic- ular space T⊥x M ⊂ Rm+k. In general, perpendicular spaces T⊥x M and T⊥y M at different points x and y of M may have common points in Rm+k. However, if M is compact, then, by the Tubular Neighborhood Theorem,5 for small ε the exponential map is a diffeomorphism onto a neighborhood of M, called a tubular neighborhood of M. In other words, for small values of ε, the exponential map identifies a neighborhood of M in Rm+k with a tube M× Dε.

1.2 Cobordisms of framed manifolds

The Pontryagin construction shows that a class in πm+kSk defines a (closed) framed manifold M uniquely up to a cobordism. In fact, there is a bijective correpsondence between framed manifolds of dimension m in Rm+k up to cobordisms and pointed maps Sm+k → Sk up to ho- motopy, where cobordisms of m-manifolds in Rm+k is an equivalence relation which we define next.

stable homotopy groups of spheres 19

Figure 1.7: A manifold with bound- ary, and the collar neighborhood of its boundary.

6 Hint to Exercise 1.3. Let f : Rm+k → Rm+k be the parallel translation that brings M0 to M1. Let j : [0, 1] → [0, 1] be a smooth function that equals 0 for t < ε, increases on the interval (ε, 1− ε) and equals 1 for t > 1− ε. Then

Ft(x) = (1− j(t))x + j(t) f (x)

is a smooth deformation of M0 to M1 where t ∈ [0, 1] and x ∈ M0, while the trace W = {(Ft(x), t)} of the deforma- tion is a cobordism from M0 to M1. The frame over W at x is constructed in two steps. First, one applies dx(Ft) to the frame of M0 at x to get k vectors at Ft(x), and, second, one projects the k vectors to T⊥x W.

Definition 1.2. For a compact subset W in the slice Rm+k × [0, 1] of the Euclidean space Rm+k+1 and for i = 0, 1, let us denote the set of points of W on the hyperplane Rm+k × {i} by ∂iW. We say that the space W is a cobordism between ∂0W and ∂1W if the complement to ∂0W ∪ ∂1W in W is a manifold in Rm+k+1, and there is some ε > 0 such that

W ∩ (

= ∂1W × (1− ε, 1].

The union ∂W of the two sets ∂iW is called the boundary of W, while W is called a manifold with boundary. Note that the boundary ∂W has a nice neighborhood in W, namely the union of ∂0W × [0, ε) and ∂1W× (1− ε, 1], called a collar neighborhood. In particular, the boundary components ∂0W and ∂1W are smooth manifolds.

For example, for a manifold M in Rm+k the manifold with boundary W = M × [0, 1] in Rm+k × [0, 1] is a cobordism between two copies of M; it is called the trivial cobordism of M. On the other hand, W =

M× [0, 1/2) is not a cobordism between M and an empty set since W is not compact.

The technical requirement that the boundary of W possesses a col- lar neighborhood is often omitted. However, the existence of collar neighborhoods simplifies proofs. For example, in the presence of col- lar neighborhoods, the cobordism relation is clearly transitive.

Suppose now that the manifold W is framed, i.e., at each point (x, t) ∈ Rm+k× [0, 1] on the manifold W there is a chosen frame τ(x,t). Since the vectors in τ(x,t) are perpendicular to T(x,t)W, the frame τ(x,t) restricted to ∂0W and ∂1W turns the two boundary components into framed manifolds in Rm+k. Furthermore, suppose that τ(x,t) = τ(x,1) if t is ε- close to 1 and τ(x,t) = τ(x,0) if t is ε-close to 0 for some ε > 0. Then we say that W is a (framed) cobordism between the framed manifolds ∂0W and ∂1W.

The framed cobordism relation is an equivalence relation. The equiva- lence class of a framed manifold M will be denoted by [M].

Exercise 1.3. Show that a framed manifold M0 is cobordant to the framed manifold M1 obtained from M0 by a parallel translation.6

Finally, we note that the set of all cobordism classes of closed framed manifolds of dimension m in Rm+k forms an abelian group. Indeed, if M0 and M1 are two framed manifolds representing cobordism classes [M0] and [M1], we can use a parallel translation to shift M0 into the left

20 elements of surgery theory

7 If M0 and M1 do not share common points in Rm+k , then we may define the sum [M0] + [M1] to be [M0 ∪ M1] with- out taking parallel translations. How- ever, if the intersection of M0 and M1 is non-empty, the subset M0 ∪M1 of Rm+k

may not be a submanifold.

Figure 1.8: The inverse to the class of the framed manifold M is the class of the re- flection of M

8 To be more precise, we add collar neighborhoods to the boundary of W, and shrink Rm+k+1 along the (m + k + 1)-st direction so that W is a subset of Rm+k × [0, 1).

Figure 1.9: The Pontryagin construction can alternatively be described as follows: for each point x ∈ M consider its open ε-disc neighborhood Dx in the perpen- dicular space T⊥x M. These discs form a neighborhood U of M. To define the map f |U represent the sphere Sk as the the disc Dk in which all points of the boundary ∂Dk are identified. The map f sends Dx isomorphically onto Dk in such a way that the frame vectors τ1, ..., τk in Dx at x are send to the standard vectors e1, ..., ek on Dk at 0. Finally the map f is extended over Sm+k by sending the com- plement to U onto the south pole of Sk .

half space L = (−∞, 0)×Rm+k−1 of Rm+k and M1 into the right half space R = (0, ∞)×Rm+k−1, and get a new framed manifold M0 tM1

in Rm+k representing the sum [M0] + [M1].7 The so-defined operation (addition) is well-defined: changing representatives in the equivalence classes [M0] and [M1] does not change the resulting class [M0] + [M1].

The addition is clearly associative and commutative. The zero is rep- resented by an empty manifold. To construct the inverse of the cobor- dism class of a framed manifold M, place M into L and rotate L in Rm+k+1

+ = Rm+k × [0, ∞) along Rm+k−1 till L turns into R. Then the framed manifold M traces a framed cobordism W from M ⊂ L to a framed manifold −M ⊂ R.8 In particular, [M] + [−M] = 0, and there- fore −M represents the negative of [M], see Figure 1.8.

To summarize, we have shown that the cobordism classes of (closed) framed manifolds of dimension m in Rm+k form an abelian group.

1.3 The Pontryagin construction

Pontryagin observed a beautiful relation between framed manifolds M and homotopy groups of spheres. To see the relation, let us identify the complement in Sn to its south pole with Rn so that 0 ∈ Rn is the north pole of Sn; the south pole ∞ will be the base point of Sn. Let U denote an ε-tubular neighborhood of a closed framed manifold M of dimension m in Rm+k. Define f |U to be the map from U to Rk ⊂ Sk

by exp(x, v) 7→ v/(ε− |v|). It takes the manifold M to the north pole of Sk and identifies each fiber Dε of the tubular neighborhood with the Euclidean space Rk ⊂ Sk. Then, define the restriction f |Sm+k \U to be the constant map to ∞ ∈ Sk. The obtained continuous map f sends the south pole of Sm+k to the south pole of Sk, and, therefore, represents an element in πm+kSk. Choosing a different value for ε leads to the same element in πm+kSk. Thus, every closed framed manifold M in Rm+k determines a homotopy class in πm+kSk.

The corresondence [M] → [ f ] is well-defined as cobordant closed framed manifolds M0 and M1 define homotopic maps f0, f1 : Sm+k → Sk. Indeed, the Pontryagin construction applied to a cobordism W ⊂ Rm+k× [0, 1] between M0 and M1 results in a homotopy Sm+k× [0, 1]→ Sk between f0 and f1.

Conversely, let f : Sm+k → Sk be a smooth representative of an element in πm+kSk. In view of the Sard theorem, by slightly perturbing the

stable homotopy groups of spheres 21

9 Let us show that dx f |T⊥x M is an iso- morphism of the perpendicular space T⊥x M and the tangent space T0Rk . To begin with, the differential dx f is surjec- tive since the north pole of Sk is a reg- ular value of f . On the other hand, the map f takes the entire manifold M to 0, and therefore the differential dx f is triv- ial on Tx M. Consequently, the restric- tion dx f |T⊥x M is surjective. Finally, the dimension of the perpendicular space T⊥x M is the same as the dimension of T0Rk , which implies that the surjective homomorphism dx f |T⊥x M is actually an isomorphism.

10 We say that a map g is homotopic to h if there is a continuous family of maps ft parametrized by t ∈ [0, 1] such that f0 = g and f1 = h.

Figure 1.10: A framed manifold M ⊂ Rm+k also belongs to Rm+k+1. Its frame in Rm+k can be completed by em+k+1 to a frame in Rm+k+1.

map f , we may assume that the north pole of Sk is a regular value of f . Then the inverse image of the north pole is a manifold M in Rm+k ⊂ Sm+k, see Figure 1.9. Furthermore, at each point x in M the differential dx f takes the perpendicular space T⊥x M isomorphically onto the tangent space T0Rk = T0Sk at the origin. 9 Thus d f composed with the canonical isomorphism T0Rk = Rk defines a frame on M. In other words M is a framed manifold. However, the framed manifold M is not uniquely determined by the class in πm+kSk; choosing a different representative f of the class leads to a different framed manifold M.

To analyze the indeterminacy, let f0 and f1 be two different smooth representatives of the same element in πm+kSk determining two framed manifolds M0 and M1. Then f0 is homotopic10 to f1 through a smooth homotopy ft : Sm+k → Sk parametrized by t ∈ [0, 1]. Define a map f : Sm+k × [0, 1]→ Sk by f (x, t) = ft(x). We may assume that the north pole of Sk is a regular value of f0, f1 and f , and that the deformation f is trivial when the time parameter is close to 0 and 1. Then, as above, the map f determines a framed manifold W = f−1(0) in Rm+k × [0, 1] with boundary. Furthermore, the boundary of W coincides with the union of M0 and M1, and the frame of W restricts to the frames of M0

and M1. Thus W is a cobordism between the framed manifolds M0

and M1.

To summarize, we have shown that the homotopy classes in πm+kSk

are in bijective correspondence with the cobordism classes of framed manifolds.

Futhermore, the sum of cobordism classes of framed manifolds under the Pontryagin construction corresponds to addition in πm+kSk. Thus, we proved the Pontryagin Theorem.

Theorem 1.4 (Pontryagin). The group of cobordism classes of framed man- ifolds Mm ⊂ Rm+k is isomorphic to the homotopy group πm+kSk.

In the remainder of this section we will describe the stability phe- nomenon, and state the Pontryagin theorem in the form that we dis- cussed in the beginning of the lecture.

A framed manifold in Rm+k is naturally a framed manifold in a bigger space Rm+k+1; in the bigger space the frame at a point x ∈ M contains the old k frame vectors in Rm+k and the additional new basis vector em+k+1. Furthermore, any two cobordant framed manifolds in Rm+k

are also cobordant as framed manifolds in Rm+k+1. Consequently, in view of the Pontryagin theorem, there is a so-called Freudenthal homo-

22 elements of surgery theory

11 Recall that πS m denotes the m-th stable

homotopy group of spheres. It is iso- morphic to the group πm+kSk for any k > m + 1.

12 A frame τ1, ..., τk at a point p deter- mines a homomorphism of vector spaces

Rk −→ Rk

ei 7→ τi .

The group GLk(R) of homomorphisms has two path compoments; the compo- nent of homomorphisms f with det f > 0 and the component of homomor- phisms f with det f < 0. Therefore, each frame {τ1, ..., τk} can be smoothly deformed either to the basis {e1, ..., ek} or to the basis {−e1, ..., ek}.

Figure 1.11: A cobordism of framed points.

morphism πm+kSk → πm+k+1Sk+1. We will see later that it is an isomor- phism provided that k > m + 1 and it is an epimorphism for k > m, see the Freudenthal Theorem 2.4. Thus there is a characterization of stable homotopy groups of spheres in terms of framed manifolds.

Corollary 1.5 (Pontryagin). The group of cobordism classes of framed man- ifolds Mm ⊂ Rm+k is isomorphic to πS

m provided that k > m + 1.11

1.4 Example: The stable group πS 0

The Pontryagin construction identifies the stable homotopy group πS 0

with the cobordism group of (closed) framed manifolds of dimension 0 in Rk for k > 1. Being compact, such a manifold consists of finitely many points p, each of which equipped with a frame, i.e., a basis for TpRk ≈ Rk. As parallel translations do not change the cobordism class of a framed manifold (see Exercise 1.3), the exact location of points p is not essential.

A frame at any point p can be smoothly deformed12 into the standard positive basis {e1, ..., ek} or the standard negative basis {−e1, e2, ..., ek}. In particular, every class in πS

0 can be represented by a union of m positively framed points and n negatively framed points for some m, n ≥ 0. We claim that the homomorphism H : πS

0 → Z given by (m, n) 7→ m− n is well-defined and it is an isomoprhism.

Indeed, a cobordism W between two framed manifolds ∂0W and ∂1W of dimension 0 is a manifold with boundary of dimension 1, i.e., a union of finitely many circles and segments. We may discard all the circles from W, and still have a cobordism from ∂0W to ∂1W. Any remaining segment in W has two boundary components p and q. If both p and q belong to the same boundary component of W, i.e., either both belong to ∂0W or ∂1W, then the signs of their frames are opposite. On the other hand if p and q belong to the different components of W, then the signs of their frames are the same. This implies that H is well- defined as a map of sets. The disjoint union of two framed manifolds (m, n) and (m′, n′) is a framed manifold (m+m′, n+n′). The equalities

H(m + m′, n + n′) = (m + m′)− (n + n′) = H(m, n) + H(m′, n′)

show, then, that the map H is actually a homomorphism.

The homomorphism H is an epimorphism since every positive integer m is the image of the cobordism class of m positively framed points,

stable homotopy groups of spheres 23

13 The homomorphism H assigns, of course, to the homotopy class of a map Sk → Sk its degree.

while every negative integer n is the image of the cobordism class of n negatively framed points. Finally, the cobordism class of m positively framed points and m negatively framed points is trivial, which implies that the homomorphism H is injective. This completes the proof that H is an isomorphism of the stable homotopy group πS

0 onto Z. 13

1.5 Further reading

We only discussed manifolds of dimension m in Rm+k. These are usu- ally defined in advanced calculus courses, e.g., see Advanced calculus of several variables by C. H. Edwards, JR. In more advanced courses manifolds are defined without referring to the ambient space Rm+k, e.g., see the textbook Differential topology by M. W. Hirsch and An introduction to differentiable manifolds and Riemannian geometry by W. M. Boothby. Early works of L. S. Pontryagin on stable homotopy groups of spheres include Homotopy classification of the mappings of an (n+2)-dimensional sphere on an n-dimensional one (1950) and Smooth manifolds and their applications in homotopy theory (1955, 1976). Pontryagin applied his construction to compute πS

1 and πS 2 . The

Thom’s generalization of the Pontryagin construction can be found in his excellent paper Quelques propriétés globales des variétés dif- férentiables (1954), which lays the foundation of the cobordism the- ory. The J-homomorphism was defined by G. W. Whitehead in On the homotopy groups of spheres and rotation groups (1942).

The homotopy theoretic calculation of π3S2 as well as πS 1 can be found

in the textbook Topology and geometry by G. E. Bredon [Br93, page 465]. It is remarkable that even before the invention of higher ho- motopy groups by E. Cech in Höherdimensionale Homotopiegrup- pen [Ce32] and W. Hurewicz in Beiträge zur Topologie der Defor- mationen [Hu35], Heinz Hopf proved in Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche [Ho30] that there are infinitely many mutually non-homotopic pointed maps f : S3 → S2. Hopf used the linking number of links f−1x and f−1y for any two regular values x, y of f to distinguish the homotopy classes of maps f . Note that if x is the north pole of S2 and y a point near x, then f−1x and f−1y play the roles of the manifold M and its frame in the Pontryagin construction.

Figure 2.1: Steve Smale (b. 1930) https://math.berkeley.edu/ smale/

Figure 2.2: Immersion vs. non- immersion.

2

Immersion Theory

A manifold placed into Rn with possible self-intersection points is called an immersed manifold. The 50s and 60s saw a rapid develop- ment of the study of immersed manifolds. Historically, the birth of the Immersion Theory is commonly associated with the works of Stephen Smale who classified immersions of spheres, and, in particular, proved a peculiar, counterintuitive statement that the standard sphere S2 in R3 can be turned inside out by a deformation through immersions. The work of Smale was later extended by Hirsch to a classification of immersions of arbitrary manifolds.

In this lecture we will give an exposition of a modern approach to classical theorems of immersion theory, due to Rourke and Sander- son. In the prerequisite section §2.1 we introduce an ambient isotopy, which is a deformation of a manifold in Rm+k. In §2.2 we review the compression technique of Rourke and Sanderson. Smale-Hirsch the- orem is one of its applications (§99.2.2). Another application of the Rourke-Sanderson technique is the Freudenthal suspension theorem (§2.3) which we have discussed in chapter 1. We will conclude the lecture with two examples: the Smale paradox (§99.2.3), and the inter- pretation of stable homotopy groups in terms of cobordism groups of orientable immersed manifolds of dimension m in Rm+1 (§99.2.4).

To put the results of the present chapter into perspective, let’s recall that according to the Pontryagin theorem, every element of the sta- ble homotopy group πm+kSk is represented by a framed manifold M in Rm+k. We would like to simplify the representing manifold M as much as possible. In the present chapter we will use a deformation (ambient isotopy) to bring the manifold M into Rm+k−1 ⊂ Rm+k when possible. In later chapters we will attempt to replace M with a sphere

immersion theory 25

1 Let’s take a look at a simple exam- ple of how a smooth vector field turns into a differential equation. Suppose that w(x, y) is a vector field on R2

with components (x2 + y, y3), while γ(t) is a curve on R2 with components (x(t), y(t)). Then the velocity vector of γ(t) has components (x(t), y(t)), and the differential equation γ(t) = w(γ(t)) corresponding to the vector field w(x, y) takes the form{

x = x2 + y y = y3

where x = x(t) and y = y(t) are func- tions in t.

2 Existence and Uniqueness Theorem for first order differential equations. Suppose that the function F is continu- ous. Then the differential equation y(t) = F(t, y(t)) with an initial value y(t0) = y0 has a unique solution on the interval [t0 − ε, t0 + ε] for some real number ε < 0.

Figure 2.3: A vector field w and the trace γ(t) of isotopy.

3 In other words, a smooth vector field is a smooth function

w : R×Rm+k → Rm+k ,

w : (t, x) 7→ wt(x).

We may think of wt as of a smooth fam- ily of vector fields parametrized by t.

by performing spherical surgeries on M.

2.1 Ambient isotopy

In this section we will make precise the somewhat vague notion of a deformation of a manifold in Rm+k. The correct term is actually an ambient isotopy. When possible, we will use an ambient isotopy to simplify a framed manifold M in Rm+k by deforming it into the Euclidean hyperspace Rm+k−1.

A smooth vector field w on Rm+k defines a first order differential equa- tion γ(t) = w(γ(t)) with smooth coefficients;1 its solution is a curve γ : R → Rm+k whose velocity vector at the moment t is precisely the vector w at the point γ(t) ∈ Rm+k. Informally, if we imagine that the space Rm+k consists of particles, then the vector field w shows the direction for the flow of the particles, see Fig. 2.3.

We will often assume that the vectors in the vector field w are bounded in length, i.e., there is a positive number l such that |w(x)| < l for all x ∈ M. By the Existence and Uniqueness Theorem2, if the vectors w(x) of the vector field w are bounded in length, then there is a unique solution γ(t) = F(x, t) for any initial condition x = γ(0). In other words, a vector field defines a one parametric deformation F of the Euclidean space Rm+k:

Ft : Rm+k → Rm+k,

Ft : x 7→ F(x, t),

with F0 being the identity map of Rm+k. It is known that the deforma- tion (or, flow) F(x, t) associated with a smooth vector field is smooth in x and t. In fact, for each moment of time t, the map Ft is a a dif- feomorphism of Rm+k, i.e., a smooth homeomorphism whose inverse is also smooth.

Note that if we do not require that the vectors w(x) in the vector field w are bounded in length, then it may happen that the flow carries a point x = γ(0) to infinity in finite time, and therefore, the position γ(t) for large t may not be well-defined.

A smooth time dependent vector field wt smoothly3 associates at each moment of time t a vector wt(x) to each x ∈ Rm+k. Again, under the assumption that all vectors wt(x) are bounded in length, there exists a

26 elements of surgery theory

4 Let v0 be a normal vector field over M, and v1 be the vector field obtained from v by projecting each vector v0(x) to the perpendicular space T⊥x M. The lin- ear deformation of v0 to v1 is the fam- ily of vector fields vt = (1 − t)v0 + tv1 parametrized by t ∈ [0, 1].

5 Hint for Exercise 2.1. Note that the dif- ferential dFt of the diffeomorphism Ft is invertible. In particular, if a vector v(x) is not in Tx M, then the vector dFt(v(x)) is not in Ty Mt, where Mt = Ft(M) and y = Ft(x). This implies that normal vec- tor fields over M flow into normal vector fields over Mt. On the other hand, the differential dFt may not preserve the an- gles, i.e., perpendicular vector fields over M may not flow into perpendicular vec- tor fields over Mt.

flow Ft : Rm+k → Rm+k along wt in the sense that for each x the curve t 7→ Ft(x) has velocity wt(Ft(x)).

The flow F of a (possibly, time dependent) vector field is an (ambient) isotopy, i.e., it is a smooth parametric family Ft of diffeomorphisms of Rm+k such that F0 is the identity diffeomorphism. In practice, an ambient isotopy is viewed as a parametric deformation of M in Rm+k, as for each moment of time t, the set Mt = Ft(M) is also a manifold placed in Rm+k. We even say that Ft is an ambient isotopy of M. However, as the word "ambient" suggests, the flow F actually deforms not only M, but the entire background space Rm+k.

A smooth (respectively, continuous) normal vector field v on a manifold M in Rm+k is a function that associates to each point x in M a vector at x not in Tx M such that the components of the vector v(x) change smoothly (respectively, continuously) with the parameter x. Project- ing each vector v(x) to the corresponding perpendicular plane T⊥x M produces a vector field that is actually perpendicular to Tx M. Every perpendicular vector field is normal. Not ever normal vector field is perpendicular, but every normal vector field can be linearly deformed to a perpendicular one.4 Exercise 2.1 shows why we prefer to work not only with perpendicular vector fields.

Exercise 2.1. Under isotopy a manifold with a normal vector field flows into a manifold with a normal vector field. On the other hand, a manifold with a perpendicular vector field may not flow into a mani- fold with a perpendicular vector field. 5

Exercise 2.2. Let M be a manifold of dimension m in Rm+k. Suppose that normal vector fields v0 and v1 over M are homotopic through normal vector fields. Show that v0 is isotopic to v1, i.e., there is an ambient isotopy Ft with t ∈ [0, 1] such that F0 = id, Ft(x) = x for all x in M, and dF1(v0) = v1.

2.2 The Global Compression Theorem

We say that a vector field v along a manifold M in Rm+k is vertical up if for every point x ∈ M the vector v(x) is a positive multiple of the last basis vector em+k; we regard Rm+k−1 × {0} ⊂ Rm+k as horizontal.

Compression Theorem 2.1 (Rourke-Sanderson). Every normal vector field v on a closed manifold M of dimension m in Rm+k with k > 1 can

immersion theory 27

Figure 2.4: A normal frame over M = S1

in R2 can not be vertical up since at the right most point, i.e., at the point (x, y) in M with maximal value of y, the verti- cal up direction is tangent to M. In R3

every vector v(x) can be rotated to the vertical up position.

Figure 2.5: A rotation of v to w.

Figure 2.6: The neighborhood U of M consists of δ-discs perpendicular to M and centered at points x in M. Over each of the discs we extend w so that it changes linearly along radial lines from w(x) at the center x of the disc to em+k at the boundary of the disc.

be made vertical up by an ambient isotopy of the manifold M and a smooth deformation of the vector field v through normal vector fields.

The condition k > 1 in the Compression theorem is important. For example, the (radial) unit perpendicular vector field v over the stan- dard circle M = S1 in R2 can not be straightened up by an ambient isotopy of M and a deformation of v through normal vector fields. We recommend that the reader proves this fact and finds a straighten- ing up deformation of v in R3 ⊃ R2 before reading the proof of the Compression theorem, see Figure 2.4.

On the other hand, in view of Exercise 2.2 we may prohibit deforma- tions of vector fields in the Compression Theorem: Under the hypoth- esis of the Compression Theorem, there is an ambient isotopy Ft with t ∈ [0, 1] such that F0 is the identity map of Rm+k and dF1(v) = em+k

over F1(M).

Proof. To begin with we deform v into the vector field of unit length perpendicular to M. Then each vector v(x) determines a point on the unit sphere Sm+k−1; namely, if we translate v(x) so that its initial point is the origin in Rm+k−1, then its terminal point specifies a point on the unit sphere Sm+k−1. In other words, the vector field v defines a so-called Gauss map M → Sm+k−1. By the Sard Theorem, almost every value in the target of the Gauss map is regular. On the other hand, if k > 1, then regular values are precisely those with empty inverse image. Thus, if k > 1, then by slightly rotating the manifold M together with the vector field, we may assume that the image of the Gauss map avoids the south pole. That is to say, the vector field v is never vertical down.

Next we observe that since M is compact, the angle between v(x) and the vertical direction −em+k is at least ε > 0. Choose a positive real number µ < ε, and rotate each vector v(x) in the plane P spanned by v(x) and ek+m in the direction vertically up till its last component is positive and v(x) has angle at least µ with the horizontal plane. Such a rotation of the vector field v can be chosen to be through normal vector fields over M. Indeed, the intersection of Tx M with the plane of rotation P is either a point or a line whose angle with the horizontal plane is at least ε; therefore we stop rotating v(x) before reaching Tx M. Let w denote the smoothing of the rotated unit vector field over M.

The vector field w can be extended over Rk+m so that the angle of w with the horizontal plane is at least µ and outside a δ-neighborhood

28 elements of surgery theory

6 Note that since the manifold M is com- pact, it lies in a ball of finite radius. On the other hand, since the angle of w with the horizontal hyperspace is at least µ, the flow along w lifts each point by at least T sin µ units at time T.

7 Hint for Exercise 2.3. To define such an isotopy, let f be a function on π(M) that assigns to a point π(x) the last co- ordinate xm+k of the point x ∈ M. As a smooth function on M, the function f admits an extension to a smooth func- tion on all Rm+k−1. Now, define a vector field w on Rm+k by w(x, y) = − f (x)em+k where (x, y) ∈ Rm+k−1 ×R. Show that the flow of the vector field w is along the direction ±em+k and brings M to π(M) in time 1.

Figure 2.7: A framed manifold M ⊂ Rm+k also belongs to Rm+k+1. Its frame in Rm+k can be augmented by em+k+1 to form a frame in Rm+k+1.

U of M the vector field w is vertical up, for some small δ. Indeed, the neighborhood U consists of δ-discs perpendicular to M and centered at points x in M. Over each of the discs we extend w so that it changes linearly along radial lines from w(x) at the center x of the disc to em+k

at the boundary of the disc. Finally, we extend w over the rest of Rm+k

by vertical up vector filed, smooth the resulting vector field w, and multiply it by a bump function with (a very big) compact support. The flow of w isotopes M to a manifold M′ outside U in finite time.6

The modified normal vector field on M′ is vertical up.

The Compression Theorem allows us in certain cases to deform M to the hyperplane Rm+k−1. Indeed, suppose that a compact manifold M is equipped with a vertical up normal vector field v. Furthermore, suppose that none of the lines in the direction v intersects M at more than one point. Then the projection π : Rm+k → Rm+k−1 along em+k

places M into the hyperplane Rm+k−1.

Exercise 2.3. Show that there is an ambient isotopy that moves all points of Rm+k in the vertical direction and brings M to π(M). 7

We say that the ambient isotopy of Exercise 2.3 compresses the manifold M to Rm+k−1.

The Compression Theorem admits various generalizations, which we discuss in the chapter of additional topics. For example, the Multi- compression theorem asserts that if a manifold M of dimension m in Rm+k is equipped with n < k linearly independent perpendicular vec- tor fields v1, ..., vn, then there is an ambient isotopy Ft that straightens the vectors v1, ..., vn up in the sense that F0 = id and dF1(vi) = ei for i = 1, ..., n.

2.3 The Freudenthal suspension theorem

Recall that the Pontryagin construction identifies the homotopy groups πm+kSk of spheres with the cobordism classes of framed manifolds M of dimension m in the Euclidean space Rm+k. Such a manifold M also lies in a bigger space Rm+k+1. Furthermore, its frame in Rm+k can be augmented with an additional vector em+k+1 to produce a frame in Rm+k+1, see Figure 2.7. In other words, each framed manifold in Rm+k may also be considered to be a framed manifold in a bigger space Rm+k+1. Furthermore, cobordant framed manifolds in Rm+k are

immersion theory 29

8 Freudenthal homomorphism (classical definition): Every map f : Sm+k → Sk

defines the suspension map Sm+k+1 → Sk+1 by taking the poles of Sm+k+1 into the respective poles of Sk+1 and taking the longitude through any point x in Sm+k homeomorphically into the longi- tude through the point f (x). Taking suspensions of representatives, gives an- other definition of the Freudenthal ho- momorphism πm+kSk → πm+1Sk+1.

9 To show surjectivity of the Freuden- thal homomorphism, we need to show that the framed manifold M ⊂ Rm+k+1

is cobordant to a manifold in Rm+k ⊂ Rm+k+1 with v being vertically up. Re- call also that isotopies of framed mani- folds as well as deformations of normal vector fields imply cobordisms.

Figure 2.8: The manifold of distinct pairs (x, y) is obtained from the manifold M× M by removing the diagonal {x = y}.

10 Note that the proof of injectivity of the Freudenthal homomorphism does not follow immediately from the proof of surjectivity. We begin with a framed manifold M in Rm+k that is zero cobor- dant in Rm+k+1 by means of a framed cobordism W. Of course, we may find a flow F that straightens up the last vector v of the framed cobordism W, and then we can compress W to Rm+k . However if F displaces M, the resulting compressed cobordism W is a cobordism to zero of a displaced manifold M, not the original manifold M.

still cobordant as framed manifolds in Rm+k+1. In view of the Pon- tryagin construction, the correspondence between framed manifolds in Rm+k and Rm+k+1 defines a so-called Freudenthal suspension homo- morphism8 of homotopy groups of spheres πm+kSk → πm+k+1Sk+1.

Theorem 2.4. The Freudenthal homomorphism is an isomorphism for k >

m + 1 and an epimorphism for k > m.

It follows that the homotopy groups of spheres πm+kSk are the same for a fixed m and k > m + 1. These are stable homotopy groups πS

m.

Proof of the Freudenthal theorem. Suppose that k > m. Recall that a nor- mal frame v1, .., vk+1 on a manifold M in Rm+k+1 is a set of k + 1 normal vector fields over M that project to a basis of T⊥x M for each point x ∈ M. In view of the Compression theorem, we may assume that the (k + 1)-st normal vector field v = vk+1 is vertical up.9

We claim that the manifold M can be rotated slightly in Rm+k+1 so that any line in the direction em+k+1 intersects M at most at one point. Indeed, for any two distinct points x and y in M, the unit vector w(x, y) in the direction x− y points to a point in Sm+k. Since the dimension of pairs of distinct points is 2m, and the dimension of the sphere Sm+k is at least 2m + 1, the set of vectors w(x, y) is of measure zero by the Sard theorem, see Figure 2.8. Therefore by slightly rotating M we may make sure that em+k−1 is not among the vectors w(x, y), which precisely means that any line in the direction em+k+1 intersects M at most at one point. Of course, after the rotation v may not be vertically up any more. However, since we may choose the rotation to be arbitrarily small, the rotated v can be deformed back to the vertical up position.

Now we may use the vector field v to compress the framed manifold M ⊂ Rm+k+1 to a framed manifold in Rm+k, see Exercise 2.3. The compression can be iterated consecutively using the vectors vk+1, vk, ... as long as the index of the vector vi under consideration satisfies i > m. This proves surjectivity of the Freudenthal homomorphism.

To show that the Freudenthal homomorphism is injective,10 suppose that a framed manifold M ⊂ Rm+k is zero cobordant after applying the Freudenthal suspension. In other words, after including M into Rm+k+1 and adding the vector field em+k+1 to its frame, the mani- fold M becomes zero cobordant, i.e., there is a framed manifold W in Rm+k+1 × [0, 1] with boundary M ⊂ Rm+k+1 × {0}. We know that the last vector field v in the frame of W restricts to em+k+1 over M. In fact,

30 elements of surgery theory

11 We can extend v vertically up over the slice Rm+k+1 × [0, ε) since, by the defi- nition of a framed cobordism, the only points of W that are in Rm+k+1 × [0, ε) are the points of the collar neighborhood U.

Figure 2.9: The framed manifold M is placed in Rm+k which is depicted as the direction to the right. It is zero cobor- dant in the horizontal space Rm+k+1. The cobordism W itself is a manifold with boundary in Rm+k+1 × [0, 1]. If we can compress W to Rm+k × [0, 1] without displacing M, then we get a cobordism to zero of the original framed manifold M in Rm+k .

we may assume that it restricts to em+k+1 over an ε-collar neighbor- hood U of M in W.

Now let us carefully examine the argument in the proof of the Com- pression Theorem. We first deform v over W to a vector field so that the last component of each vector v(x) is positive. We may assume that during the deformation v stays vertically up over the collar neigh- borhood U, see Figure 2.9. Next we extend v to a vector field on Rm+k+1 × [0, 1]. We can extend v as in the Compression Theorem so that v is vertically up outside a neighborhood of W. In addition we may assume that v is vertically up over the slices Rm+k+1 × [0, ε) and Rm+k+1× (1− ε, 1] whose union we will denote by V.11 Then the flow F along the extended vector field is well-defined as the condition that v ≡ em+k+1 over V prevents the points of Rm+k+1 × [0, 1] from flow- ing out of the region of definition of F. The flow F takes W outside its neighborhood in a finite time T and therefore straightens up the vector field v. Note that all points in U travel the distance T in the direction v = em+k+1. Therefore, if we postcompose the ambient isotopy along v with translation along −Tem+k+1, then we get a framed cobordism W ′

bounding the original manifold M and such that v is vertical up.

Finally, when k > m + 1, we can use the same argument as in the proof of surjectivity to show that W ′ compresses to a cobordism of M in Rm+k × [0, 1]. Thus, a framed manifold M ⊂ Rm+k is cobordant to zero after the Freudenthal suspension only if it is cobordant to zero itself.

2.4 Further reading

We borrowed the proof of the Compression Theorem from the original paper The compression theorem I [RS01] of C. Rourke and B. Sander- son. The theorem has many interesting applications some of which are explained in The compression theorem II: directed embeddings [RS01a] and The compression theorem III: applications [RS03]. We gave one more application here: the Freudenthal theorem.

The original proof of Freudenthal suspension theorem in Über die Klassen der Sphärenabbildungen. I. Große Dimensionen [Fr38] was deemed hard. The modern homotopy theoretic proof is based on the Blackers-Massey excision theorem proved in The homotopy groups of a triad [BM51]. Recall that in general the Freudenthal homomorphism E is not an isomorphism below the stable range. In the paper On the

immersion theory 31

12 To find the EHP exact sequence, Whitehead observed that E is the ho- momorphism induced by an embedding Sn → Sn+1, and therefore E fits the ho- motopy long exact sequence of the pair (Sn+1, Sn).

Freudenthal theorem [Wh53], Whitehead fitted E into the so-called EHP sequence12

· · · → πqSn E−→ πq+1Sn+1 H−→ πq−1S2n−1 P−→ πq−1Sn → · · · .

Promptly James proved in On the iterated suspension [J54] the ex- istence of a similar long exact sequence for an iterated Freudenthal homomorphism Em : πqSn → πq+mSn+m. Its geometric interpretation in terms of Pontryagin construction can be found in the paper Geo- metric interpretations of the generalized Hopf invariant [KS77] by Koschorke and Sanderson.

The original proof of the Smale theorem appeared in his papers A classification of immersions of the two-sphere and The classifica- tion of immersions of spheres in Euclidean spaces, [Sm58, Sm59]. Smale’s theorems were generalized by Hirsch in the paper Immer- sions of manifolds[Hi59]. It was observed later that not only immer- sions can be replaced with their formal counterparts. A number of remarkable discoveries culminated in the so-called homotopy princi- ple (h-principle, for short). Besides the classical reference of Partial differential relations [Gr86] by Gromov, we recommend the Introduc- tion to the h-principle [EM02] by Eliashberg and Mishachev.

Figure 3.1: Marston Morse (1892–1977)

3

Spherical surgeries

Under the Pontryagin construction, every element of the homotopy group πm+kSk is identified with the cobordism class of framed man- ifolds of dimension m in Rm+k. The choice of a representing framed manifold in the cobordism class is far from being unique, and it is our goal to find a simple representative. Our strategy is to begin with an arbitrary (a priory complicated) representative and then use a framed cobordism to simplify it as much as possible.

A general cobordism on a manifold W0 could be overly complex: it is hard to describe such a cobordism and its action on the homotopy groups πiW0. However, we will see that every non-trivial cobordism breaks into a composition of elementary, so-called spherical cobordisms. A spherical cobordism on a manifold W0 performs a spherical surgery on W0. A spherical surgery is relatively easy to describe in terms of (locally flat) topological submanifolds of Rm+k which are almost every- where smooth. We will discuss a general Cairns-Whitehead technique of smoothing topological manifolds, and apply it to smooth corners and more complicated singularities produced by an embedded spher- ical surgery.

Next, we will learn how to efficiently encode a spherical surgery in terms of a base of surgery. We will determine how a spherical surgery on a manifold W0 changes its homotopy groups. As a demonstration of the introduced surgery technique, we will calculate the homotopy group π3S2 as well as the stable homotopy groups πS

1 and πS 2 .

spherical surgeries 33

1 Recall, the differential dx f of the height function linearly projects the space TxW ⊂ Rm+k ×R to the last com- ponent.

Figure 3.2: A cobordism W between W0 and W1 with two critical points. The tan- gent planes at critical points are horizon- tal.

Figure 3.3: After slightly perturbing W from Figure 3.2, the height function has only Morse critical points.

3.1 Surgery and cobordisms

Trivial cobordisms

Recall that a cobordism of framed manifolds of dimension m in Rm+k

is a framed closed manifold W of dimension m+ 1 in Rm+k× [0, 1]. The projection f of W onto the last coordinate is a so-called hight function. We say that a point x ∈ W is regular if dx f is surjective, and critical otherwise.1 Geometrically, the critical points of a height function are those at which the tangent plane is horizontal, see Figure 3.2.

There is a simple coordinate description of regular and critical points. Namely, by the Rank theorem, in a neighborhood of a regular point, there are coordinates (x1, ..., xm, xm+1) on W such that f (x) = xm+1. On the other hand, after slightly perturbing W, near each critical point there are coordinates such that

f (x1, ..., xm+1) = −x2 1 − · · · − x2

i + x2 i+1 + · · · x2

m+1,

see Figure 3.3. The latter statement—which we will not prove here—is known as the Morse lemma, while the perturbed function is called a Morse function.

The integer i in the coordinate representation of f near a critical point p may be different for different critical points. It is called the index of the critical point. The index of a critical point p does not depend on the choice of a coordinate chart about p. Note that if p is a critical point of f of index i, then p is also a critical point of − f of index j = m + 1− i.

For example, in the Figure below the cobordism has two Morse critical points, the points p1 and p2. At the point p1 the height function f has a local minimum; in a neighborhood of p1 in appropriate coordinates f can be written as f (x1, x2) = x2

1 + x2 2. The index of the critical point p1

is 0. The point p2 is a saddle point. In its neighborhood in appropriate coordinates the height function has the form f (x1, x2) = −x2

1 + x2 2. In

particular, the index of p2 is 1.

According to Lemma 3.1, as t changes from 0 to 1, the regular levels Wt = f−1(t), i.e., levels with no critical points of f , change by isotopy.

34 elements of surgery theory

Figure 3.4: The vertically up vector field em+k+1 and its projection v to W.

2 Hint to Exercise 3.2 We have defined the map

: W0 × [0, 1]→W

in the proof of Lemma 3.1 by (x, t) = Ft(x). The flow F carries points along the scaled vector field w. Since the last component of the scaled vector field w is 1, the last component of each point in- creases with velocity 1. Thus, in time t, the map Ft lifts each point of W0 to the hight t. In other words, (W0 × {t}) ⊂ Wt. Since the flow F−1 along the nega- tive of the scaled vector field w is smooth and brings Wt to W0 we deduce that |W0 × {t} is a diffeomorphism. In par- ticular, identifies W0 × {t} with Wt. Since is a homeomorphism and d is of full rank, it follows that is a diffeo- morphism, e.g., see [Lee13].

Lemma 3.1. Suppose that f has no critical points. Then the cobordism W is trivial, that is there is a diffeomorphism : W0 × [0, 1] → W that identifies W0 × {t} with the t-th level Wt of f for all t.

Sketch of the proof. Consider the vector field em+k+1 in the slice Rm+k× [0, 1] of a Euclidean space. Since the hight function f has no critical points, over W the projection v of em+k+1 to W is never horizontal, i.e., the last component vm+k+1 of the vector field v is never zero. If we scalar multiply v by the smooth function 1/vm+k+1, then we obtain a vector field w over W with last component 1. The flow F of the manifold W0 along the scaled vector field w carries W0 along W and brings it to Wt at the time t. In fact, F defines a desired diffeomorphism by taking a point (x, t) to Ft(x).

Note that in the proof of Lemma 3.1 we need to scale the vector field v in order to make sure that for any point x ∈ ∂0W the last coordinate of the curve t 7→ Ft(x) increases with speed 1 so that the flow Ft indeed brings W0 to the t-th level at the time t.

Exercise 3.2. Show that the map is a diffeomorphism that identifies W0 × {t} with Wt. In particular, show that −1 is smooth.2

Spherical cobordisms

In view of Lemma 3.1 as t increases from 0 to 1 the level Wt essen- tially changes only when t passes a critical value, i.e, the value f (x) of a critical point x. In appropriate coordinates, the gradient of f in a neighborhood of a Morse critical point is

d f (x1, ..., xm+1) = 2(−x1, ...,−xi, xi+1, ..., xm+1),

spherical surgeries 35

j ,

(−2x1, ...,−2xi , 2xi+1 + 2xm+1),

or, in other words, d f (x) = 0 only if x = 0.

Figure 3.5: By perturbing W, we may as- sume that each level Wt contains at most one critical point of the hight function.

4 Hint for Exercise 3.3. Suppose that the hight function on the cobordism W has n critical points p1, ..., pn. We may stretch Rm+k × [0, 1] along the (m + k + 1)-st co- ordinate in such a way that for the new hight function f on W we have f (pi) ∈ (i − 1, i). Next we may modify W near Wi by ambient isotopy so that each level Wt near Wi for i = 1, ..., n− 1 is obtained from Wi by a vertical translation. Then W is a composition of spherical cobor- disms, and it is diffeomorphic to the original cobordism.

5 Near the critical point of index i, in Morse coordinate neighborhood, the core disc of a spherical surgery consists of the points (x1, ..., xi , 0, ..., 0), while the belt disc consists of the points (0, ..., 0, xi+1, ..., xm+1).

which means that the set of critical points of a Morse function is dis- crete.3 It is even finite, since W is compact. Furthermore, by slightly perturbing W, we may always assume that each level Wt contains at most one critical point of f , see Figure 3.5.

Thus, every non-trivial cobordism is a composition of spherical cobor- disms W whose height function has at most one critical point. Under a spherical cobordism, the manifold ∂0W = W0 is modified into a mani- fold ∂1W = W1 by the so-called spherical surgery.

Exercise 3.3. Let W be a non-trivial cobordism such that each level Wt

contains at most one critical point of its hight function. Show that W is diffeomorphic to a composition of spherical cobordisms.4

Let’s describe the result of a spherical surgery. The projection v of the vertical up vector field over W to W is a smooth vector field. It is zero only at the critical point p of the height function since only at the critical point p the tangent space TpW is horizontal. Let Fv denote the flow along the projected vector field v. The set of points x in W with Fv

t (x) → p as t → ∞ is called the core disc of the spherical surgery, see Figure 3.6. Similarly, the set of points x with Fv

t (x) → p as t → −∞ is called the belt disc of the spherical surgery.5 We will see shortly that the core and belt discs are indeed discs Di of dimension i and Dj of dimension j = m = 1− i respectively. The boundary ∂Di is called the attaching sphere of the surgery, while ∂Dj is called the belt sphere. The attaching sphere Si−1 is a sphere on the level W0, while the belt sphere Sj−1 is one on the level W1.

All points on W0 \ Si−1 are carried by the flow Fv t (x) to points on W1 \

Sj−1. The flow of the vector field −v defines the inverse map showing that the manifold W0 \ Si−1 is diffeomorphic to W1 \ Sj−1. In fact, we may choose neighborhoods hi and hj of the core and belt spheres so that the flow defines a diffeomorphism between W0 \ hi and W1 \ hj. Thus, we determined a geometric description of a spherical surgery: up to isotopy the manifold W1 is obtained from W0 by replacing hi

with hj, see Figure 3.7.

To identify hi and hj, we may assume that the unique critical point p of the hight function is on the level W1/2. Since f has no other critical points, we already know that all levels Wt for t ∈ [0, 1/2− ε]

are mutually diffeomorphic. The same is true for levels Wt with t ∈ [1/2+ ε, 1]. In fact, we may remove f−1[0, 1/2− ε) and f−1(1/2+ ε, 1] from the cobordism W without changing its diffeomorphism type. To simplify notation, we will denote the parts of the core and the cocore

36 elements of surgery theory

Figure 3.6: The belt disc Dj is above the critical point (red), the core disc Di is be- low the critical point (red). The traces of the flow F outside the belt and core disc (blue) run from W0 to W1.

Figure 3.7: The handles hi and hj are in green. The flow F carries W0 \ hi to W1 \ hj.

Figure 3.8: The manifolds W1/2−ε and W1/2+ε.

discs Di and Dj remaining in W by the same symbols, and call their boundaries the attaching and belt spheres.

The manifolds W1/2+ε and W1/2−ε in a neighborhood of the critical point p are depicted in Figure 3.8. The integral curves of the flow Ft

are perpendicular to the level sets Wt, see Figure 3.9. In particular, the core disc Di is a disc in the plane (x1, ..., xi, 0, ..., 0), while the cocore disc Dj is a disc in the plane (0, ..., 0, xi+1, ..., xm+1), see Figure 3.8. The neighborhood hi of the attaching sphere Si−1 in W1/2−ε is diffeomor- phic to Si−1×Dj. The rest of W1/2−ε flows under Ft to the complement in W1/2+ε to a neighborhood hj of the belt sphere Sj−1. The neighbor- hood hj is diffeomorphic to Di × Sj−1, and W1/2+ε is obtained from W1/2−ε by removing hi, transferring the remaining part by Ft to the level 1/2 + ε, and, finally, attaching hj.

Example 3.4. There is a spherical surgery of index 1 on a sphere S2

that results in a torus, see Figure 3.10. A spherical surgery of index 1 removes a neighborhood h1 = S0 × D2 of an attaching sphere S0, and then attaches a handle h2 = D1 × S1, which is a cylinder. If we turn the corresponding cobordism W upside down, then we obtain a cobordism corresponding to a spherical surgery turning a torus into a sphere S2, see Figure 3.11. Now the spherical surgery is of index 2. It removes a neighborhood h2 = S1 × D2 of an attaching sphere S1, and then attaches a handle h1 = D2 × S0.

Conversely, let W0 be a manifold of dimension m in Rm+k, and let Di

be a disc of dimension i in Rm+k together with j linearly independent perpendicular vector fields v1, ..., vj over Di such that ∂Di is a sphere in W0 while all vector fields v`|∂Di are tangent to W0. Suppose that Di

intersects the manifold W0 only along ∂Di. Then, we say that the disc Di together with the j perpendicular vector fields is a base of spherical surgery. We call Di the core disc.

In view of the exponential map, the disc Di together with the perpen- dicular vector fields v` are often replaced with a so-called embedded i-handle Hi ⊂ Rm+k which is a thickening Di × Dj of the core disc Di

in Rm+k, see Figure 3.12. In fact, given a base of surgery (Di, v1, ..., vj), we may choose an embedding ψ : Di × Dj → Rm+k of the i-handle Hi so that ψ(Di × {0}) is the core disc, the i-handle Hi intersects W0

along ∂Di × Dj, and ψ({0} × ei) = εvi for some small real number ε

and every basis vector ei in the standard unit disc Dj ⊂ Rj.

It can be shown that in the presence of a base of spherical surgery, there exists a spherical surgery of W0 to a manifold W1 such that hi

spherical surgeries 37

Figure 3.9: The curves of the flow Ft.

Figure 3.10: A surgery turning a 2- sphere into a torus.

Figure 3.11: A surgery turning a torus into a 2-sphere.

is given by ∂Hi ∩W0, while W1 is obtained from W0 \ hi by taking its union with hj = ∂Hi \ hi and smoothing the corners. To construct the corresponding spherical cobordism W, we begin with a collar neigh- borhood W0 × [0, 1/2] in Rm+k × [0, 1/2], and attach to it a copy of Hi

in Rm+k × {1/2}, see Figure 3.13. The interior of the obtained space is a manifold with corners. Its boundary is a copy of a manifold with corners W1. Finally, we attach to the constructed space the final part W1 × [1/2, 1] in Rm+k × [1/2, 1]. The resulting cobordism

W = (W0 × [0, 1/2]) ∪ Hi ∪ (W1 × [1/2, 1])

is a so-called topological manifold. We can smooth it, and turn W into a smooth cobordism between smooth manifolds W0 and W1.

Example 3.5. The spherical surgery in Figure 3.10 corresponds to the base of surgery in Figure 3.14. Note that the base of surgery of index n consists of a disc Dn of dimension n together with dim W0 − dim ∂Dn

perpendicular vector fields along Dn. The base of surgery of index 1 in Figure 3.14 consists of the disc of dimension 1 as well as 2 perpendicu- lar vector fields. In Figure 3.15, the surgery is of index 2. Therefore the base of surgery consists of a disc of dimension 2 as well as 1 normal vector field.

3.2 Framed base of surgery

In the previous section we have seen how a base of surgery defines a spherical surgery. However, we are still to answer the question, "Given a base of surgery on a framed manifold, does it correspond to a framed surgery?" Recall that we are interested in framed surgeries rather than surgeries as the Pontryagin construction identifies homotopies of maps of spheres with framed cobordisms rather than cobordisms.

We will see that the answer to the question is negative in general: in order to define a framed surgery, we need a base of framed surgery rather than a base of surgery.

We will say that Rm+k = Rm+k×{0} is the horizontal space in Rm+k+1

while the last factor in Rm+k+1 will be referred to as the vertical space. Let W0 be a manifold of dimension m in the horizontal space Rm+k × {0} framed by vector fields τ1, ..., τk. Let Di be a disc in Rm+k ×